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Standards\$for\$<strong>Math</strong>ematical\$Practices\$The Standards for <strong>Math</strong>ematical Practice describe varieties of expertise that mathematicseducators at all levels should seek to develop in their students. These practices rest onimportant “processes and proficiencies” with longstanding importance in mathematicseducation. The first of these are the NCTM process standards of problem solving,reasoning and proof, communication, representation, and connections. The second are thestrands of mathematical proficiency specified in the National Research Council’s reportAdding It Up: adaptive reasoning, strategic competence, conceptual understanding(comprehension of mathematical concepts, operations and relations), procedural fluency(skill in carrying out procedures flexibly, accurately, efficiently and appropriately), andproductive disposition (habitual inclination to see mathematics as sensible, useful, andworthwhile, coupled with a belief in diligence and one’s own efficacy).1. Make sense of problems and persevere in solving them.<strong>Math</strong>ematically proficient students start by explaining to themselves the meaning of aproblem and looking for entry points to its solution. They analyze givens, constraints,relationships, and goals. They make conjectures about the form and meaning of thesolution and plan a solution pathway rather than simply jumping into a solution attempt.They consider analogous problems, and try special cases and simpler forms of theoriginal problem in order to gain insight into its solution. They monitor and evaluate theirprogress and change course if necessary. Older students might, depending on the contextof the problem, transform algebraic expressions or change the viewing window on theirgraphing calculator to get the information they need. <strong>Math</strong>ematically proficient studentscan explain correspondences between equations, verbal descriptions, tables, and graphsor draw diagrams of important features and relationships, graph data, and search forregularity or trends. Younger students might rely on using concrete objects or pictures tohelp conceptualize and solve a problem. <strong>Math</strong>ematically proficient students check theiranswers to problems using a different method, and they continually ask themselves,“Does this make sense?” They can understand the approaches of others to solvingcomplex problems and identify correspondences between different approaches.3[PLSD]

Standards\$for\$<strong>Math</strong>ematical\$Practices\$2. Reason abstractly and quantitatively.<strong>Math</strong>ematically proficient students make sense of quantities and their relationships inproblem situations. They bring two complementary abilities to bear on problemsinvolving quantitative relationships: the ability to decontextualize—to abstract a givensituation and represent it symbolically and manipulate the representing symbols as if theyhave a life of their own, without necessarily attending to their referents—and the abilityto contextualize, to pause as needed during the manipulation process in order to probeinto the referents for the symbols involved. Quantitative reasoning entails habits ofcreating a coherent representation of the problem at hand; considering the units involved;attending to the meaning of quantities, not just how to compute them; and knowing andflexibly using different properties of operations and objects.3. Construct viable arguments and critique the reasoning of others.<strong>Math</strong>ematically proficient students understand and use stated assumptions, definitions,and previously established results in constructing arguments. They make conjectures andbuild a logical progression of statements to explore the truth of their conjectures. Theyare able to analyze situations by breaking them into cases, and can recognize and usecounterexamples. They justify their conclusions, communicate them to others, andrespond to the arguments of others. They reason inductively about data, making plausiblearguments that take into account the context from which the data arose. <strong>Math</strong>ematicallyproficient students are also able to compare the effectiveness of two plausible arguments,distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw inan argument—explain what it is. Elementary students can construct arguments usingconcrete referents such as objects, drawings, diagrams, and actions. Such arguments canmake sense and be correct, even though they are not generalized or made formal untillater grades. Later, students learn to determine domains to which an argument applies.Students at all grades can listen or read the arguments of others, decide whether theymake sense, and ask useful questions to clarify or improve the arguments.4[PLSD]

Standards\$for\$<strong>Math</strong>ematical\$Practices\$4. Model with mathematics.<strong>Math</strong>ematically proficient students can apply the mathematics they know to solveproblems arising in everyday life, society, and the workplace. In early grades, this mightbe as simple as writing an addition equation to describe a situation. In middle grades, astudent might apply proportional reasoning to plan a school event or analyze a problem inthe community. By high school, a student might use geometry to solve a design problemor use a function to describe how one quantity of interest depends on another.<strong>Math</strong>ematically proficient students who can apply what they know are comfortablemaking assumptions and approximations to simplify a complicated situation, realizingthat these may need revision later. They are able to identify important quantities in apractical situation and map their relationships using such tools as diagrams, two-waytables, graphs, flowcharts and formulas. They can analyze those relationshipsmathematically to draw conclusions. They routinely interpret their mathematical resultsin the context of the situation and reflect on whether the results make sense, possiblyimproving the model if it has not served its purpose.5. Use appropriate tools strategically.<strong>Math</strong>ematically proficient students consider the available tools when solving amathematical problem. These tools might include pencil and paper, concrete models, aruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statisticalpackage, or dynamic geometry software. Proficient students are sufficiently familiar withtools appropriate for their grade or course to make sound decisions about when each ofthese tools might be helpful, recognizing both the insight to be gained and theirlimitations. For example, mathematically proficient high school students analyze graphsof functions and solutions generated using a graphing calculator. They detect possibleerrors by strategically using estimation and other mathematical knowledge. When makingmathematical models, they know that technology can enable them to visualize the resultsof varying assumptions, explore consequences, and compare predictions with data.<strong>Math</strong>ematically proficient students at various grade levels are able to identify relevantexternal mathematical resources, such as digital content located on a website, and usethem to pose or solve problems. They are able to use technological tools to explore anddeepen their understanding of concepts.5[PLSD]

Standards\$for\$<strong>Math</strong>ematical\$Practices\$6. Attend to precision.<strong>Math</strong>ematically proficient students try to communicate precisely to others. They try touse clear definitions in discussion with others and in their own reasoning. They state themeaning of the symbols they choose, including using the equal sign consistently andappropriately. They are careful about specifying units of measure, and labeling axes toclarify the correspondence with quantities in a problem. They calculate accurately andefficiently, express numerical answers with a degree of precision appropriate for theproblem context. In the elementary grades, students give carefully formulatedexplanations to each other. By the time they reach high school they have learned toexamine claims and make explicit use of definitions.7. Look for and make use of structure.<strong>Math</strong>ematically proficient students look closely to discern a pattern or structure. Youngstudents, for example, might notice that three and seven more is the same amount asseven and three more, or they may sort a collection of shapes according to how manysides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7× 3, in preparation for learning about the distributive property. In the expression x2 + 9x+ 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize thesignificance of an existing line in a geometric figure and can use the strategy of drawingan auxiliary line for solving problems. They also can step back for an overview and shiftperspective. They can see complicated things, such as some algebraic expressions, assingle objects or as being composed of several objects. For example, they can see 5 – 3(x– y)2 as 5 minus a positive number times a square and use that to realize that its valuecannot be more than 5 for any real numbers x and y.8. Look for and express regularity in repeated reasoning.<strong>Math</strong>ematically proficient students notice if calculations are repeated, and look both forgeneral methods and for shortcuts. Upper elementary students might notice whendividing 25 by 11 that they are repeating the same calculations overand over again, and conclude they have a repeating decimal. By paying attention to thecalculation of slope as they repeatedly check whether points are on the line through (1, 2)with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2+ x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sumof a geometric series. As they work to solve a problem, mathematically proficientstudents maintain oversight of the process, while attending to the details. Theycontinually evaluate the reasonableness of their intermediate results.6[PLSD]

5 th \$<strong>Grade</strong>\$<strong>Math</strong>\$Framework\$UNIT 1Approximately 8WeeksUNIT 2Approximately 3WeeksUNIT 3Approximately 8WeeksUNIT 4Approximately 8WeeksUNIT 5Approximately 2WeeksNumber andOperations inBase TenUnderstand theplace valuesystem(5.NBT.1,2,3,4)Performoperations withmulti digit wholenumbers and withdecimals tohundredths(5.NBT.5,6,7)Operations andAlgebraicThinkingWrite andinterpretnumericalexpressions(5.OA.1,2)Analyze patternsand relationships(5.OA.3)Measurementand DataUnderstandconcepts ofvolume andrelate volume tomultiplicationand to addition.(5.MD.3,4,5)Number andOperationsFractionsApply andextend previousunderstandingsof multiplicationand division tomultiply anddivide fractions.(5.NF.3,4,5,6,7,)GeometryClassify twodimensionalfigures intocategories basedon theirproperties.(5.G.3,4)Measurementand DataConvert likemeasurementunits within agivenmeasurementsystem(5.MD.1)GeometryGraph points onthe coordinateplane to solvereal world andmathematicalproblems(5.G.1,2)Number andOperationsFractionsUse equivalentfractions as astrategy to addand subtractfractions.(5.NF.1,2)Measurementand DataRepresent andinterpret data.(5.MD.2)Remaining weeks of school year will be utilized for review and 6 th gradeintroductions.7[PLSD]

5 th \$<strong>Grade</strong>\$Essential\$Outcomes\$Number and Operations – Fractions1. Students will be able to add and subtract fractions with unlike denominators(including mixed numbers) and solve word problems involving addition andsubtraction of fractions. (5.NF.1 and 2)2. Students will interpret a fraction as division of the numerator by the denominator(a/b = a ÷ b). (5.NF.3)3. Students will apply and extend previous understandings of multiplication tomultiply a fraction or whole number by a fraction. (5.NF.4b)b. Find the area of a rectangle with fractional side lengths by tiling it with unitsquares of the appropriate unit fraction side lengths, and show that the area isthe same as would be found by multiplying the side lengths. Multiply fractionalside lengths to find areas of rectangles, and represent fraction products asrectangular areas.4. Students will interpret multiplication as scaling (resizing) by comparing the sizeof a product to the size of one factor on the basis of the size of the other factor,without performing the indicated multiplication and then explain your answer.(5.NF.5 a,b)5. Students will solve real world problems involving multiplication of fractions andmixed numbers, e.g., by using visual fraction models or equations to represent theproblem. (5.NF.6)6. Students will use visual fraction models to divide unit fractions by whole numbersand whole numbers by unit fractions in real world problems. (5.NF.7 a,b,c)Numbers and Operations in Base Ten7. Recognize that in a multi-digit number, a digit in one place represents 10 times asmuch as it represents in the place to its right and 1/10 of what it represents in theplace to its left. (5.NBT.1)8. Explain patterns in the number of zeros of the product when multiplying a numberby powers of 10, and explain patterns in the placement of the decimal point whena decimal is multiplied or divided by a power of 10. Use whole-number exponentsto denote powers of 10. (5.NBT.2)8[PLSD]

5 th \$<strong>Grade</strong>\$Essential\$Outcomes\$9. Read, write, and compare decimals to thousandths. (5.NBT.3 a,b)a. Read and write decimals to thousandths using base-ten numerals, numbernames, and expanded form,e.g., 347.392 = 3 × _100 + 4 × _10 + 7 × _1 + 3 × _(1/10) + 9 × _(1/100) + 2 × _(1/1000).b. Compare two decimals to thousandths based on meanings of the digits in eachplace, using >, =, and < symbols to record the results of comparisons.10. Fluently multiply multi-digit whole numbers using the standard algorithm.(5.NBT.5)11. Find whole-number quotients of whole numbers with up to four-digit dividendsand two-digit divisors, using strategies based on place value, the properties ofoperations, and/or the relationship between multiplication and division. Illustrateand explain the calculation by using equations, rectangular arrays, and/or areamodels. (5.NBT.6)12. Add, subtract, multiply, and divide decimals to hundredths, using concrete modelsor drawings and strategies based on place value, properties of operations, and/orthe relationship between addition and subtraction; relate the strategy to a writtenmethod and explain the reasoning used. (5.NBT.7)Measurement and Data13. Students will make a line plot to display a data set of measurements in fractions ofa unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problemsinvolving information presented in line plots. (5.MD.2)14. Students will apply the formulas V = l × w × h and V = b × h for rectangularprisms to find volumes of right rectangular prisms with whole-number edgelengths in the context of solving real world and mathematical problems.(5.MD.5a)15. Students will recognize volume as additive. Find volumes of solid figurescomposed of two non-overlapping right rectangular prisms by adding the volumesof the non-overlapping parts, applying this technique to solve real worldproblems. (5.MD.5c)9[PLSD]

Unit 1: 5 th <strong>Grade</strong> Quarter 1 Approximately8 weeksDomain: Number and Operations in Base TenStandards5.NBT.1 Recognize that in a multi-digit number, a digit in one placerepresents 10 times as much as it represents in the place to its right and 1/10of what it represents in the place to its left.5.NBT.2 Explain patterns in the number of zeros of the product whenmultiplying a number by powers of 10, and explain patterns in the placementof the decimal point when a decimal is multiplied or divided by a power of10. Use whole-number exponents to denote powers of 10.5.NBT.3 Read, write, and compare decimals to thousandths.a. Read and write decimals to thousandths using base-ten numerals, numbernames, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 ×(1/10) + 9 × (1/100) + 2 × (1/1000).b. Compare two decimals to thousandths based on meanings of the digits ineach place, using >, =, and < symbols to record the results of comparisons.5.NBT.4 Use place value understanding to round decimals to any place.5.NBT.5. Fluently multiply multi-digit whole numbers using the standardalgorithm.5.NBT.6. Find whole-number quotients of whole numbers with up to fourdigitdividends and two-digit divisors, using strategies based on place value,the properties of operations, and/or the relationship between multiplicationand division. Illustrate and explain the calculation by using equations,rectangular arrays, and/or area models.5.NBT .7. Add, subtract, multiply, and divide decimals to hundredths, usingconcrete models or drawings and strategies based on place value, propertiesof operations, and/or the relationship between addition and subtraction; relatethe strategy to a written method and explain the reasoning used.5.MD.1 Convert among different-sized standard measurement units within agiven measurement system (e.g., convert 5 cm to 0.05 m), and use these10[PLSD]

5.NBT.5-1i) Tasks do not explicitly assess fluency.ii) The given factors are such as to require an efficient/standard algorithm(e.g., 726 × 48) . Factors in the task do not suggest any obvious ad hoc ormental strategy (as would be present for example in a case such as 725×40).iii) Tasks do not have a context.iv) For purposes of assessment, the possibilities are 2-digit × 3- digit.5.NBT.6i) Students need not use technical terms such as commutative, associative,distributive, or property.ii) Tasks do not have a context.5.NBT.7i) Students need not use technical terms such as commutative, associative,distributive, or property.ii) Tasks do not have a context.PARCCEOYClarification5.NBT.1Recognize that in a multi-digit number, a digit in one place represents 10times as much as it represents in the place to its right and 1/10 of what itrepresents in the place to its left.i) Tasks have “thin context” or no context.ii) Tasks involve the decimal point in a substantial way (e.g., by involving acomparison of a tenths digit to a thousandths digit or a tenths digit to a tensdigit).5.NBT.2-2Use whole-number exponents to denote powers of 10.i) The explanation aspect of 5.NBT.2 is not assessed here.5.NBT.3a12[PLSD]

iv) The subtrahend and minuend are each greater than or equal to 0.01 andless than or equal to 99.99. Positive differences only. (Every includedsubtraction problem is an unknown-addend problem included in5.NBT.7-3Multiply tenths with tenths or tenths with hundredths, using concrete modelsor drawings and strategies based on place value, properties of operations,and/or the relationship between addition and subtraction; relate the strategy toa written method and explain the reasoning used.i) Tasks do not have a context.ii) Only the product is required; explanations are not assessed here.iii) Prompts may include visual models, but prompts must also present thefactors as numbers, and the answer sought is a number, not a picture.iv) Each factor is greater than or equal to 0.01 and less than or equal to 99.99v) The product must not have any non-zero digits beyond the thousandthsplace.(Forexample,1.67×0.34=0.5678 is excluded because the product has an8 beyond the thousandths place; cf. 5.NBT.3 and see p. 17 of Progression forNumber and Operations in Base Ten.)vi) Problems are 2-digit × 2-digit or 1-digit by 3- or 4-digit. (For example,7.8×5.3 or 0.3×18.24 .)vii) 20% of cases involve a whole number – either the product is a wholenumber, or else one factor is a whole number presented without a decimalpoint. (Both factors cannot both be whole numbers.)5.NBT.7-4Divide in problems involving tenths and/or hundredths, using concretemodels or drawings and strategies based on place value, properties ofoperations, and/or the relationship between addition and subtraction; relatethe strategy to a written method and explain the reasoning used.i) Tasks do not have a context.ii) Only the quotient is required; explanations are not assessed here.iii) Prompts may include visual models, but prompts must also present thedividend and divisor as numbers, and the answer sought is a number, not apicture.iv) Divisors are of the form XY, X0, X, X.Y, 0.XY, 0.X, or 0.0X (cf.5.NBT.6) where X and Y represent non-zero digits. Dividends are of the formXYZ.W, XY0.X, X00.Y, XY.Z, X0.Y, X.YZ, X.Y, X.0Y, 0.XY, or 0.0X,where X, Y, Z, and W represent non- zero digits. [(Also add XY, X0, and15[PLSD]

X.)]v) Quotients are either whole numbers or else decimals terminating at thetenths or hundredths place. (Every included division problem is an unknownfactorproblem included in 5.NBT.7-3.)vi) 20% of cases involve a whole number – either the quotient is a wholenumber, or the dividend is a whole number presented without a decimal point,or the divisor is a whole number presented without a decimal point. (If thequotient is a whole number, then neither the divisor nor the dividend can be awhole number.)5.NBT.Int.1Perform exact or approximate multiplications and/or divisions that are bestdone mentally by applying concepts of place value, rather than by applyingmulti-digit algorithms or written strategies.i) Tasks have no context.ii) See ITN Appendix F, section A, “Illustrations of Innovative TaskCharacteristics,” subsection 4, “Integrative tasks with machine scoring ofresponses entered by computer interface,” subsection “Illustrations at thedomain level.”5.Int.1Solve one-step word problems involving multiplying multi-digit wholenumbers.i) The given factors are such as to require an efficient/standard algorithm(e.g., 726 ×"4871 ). Factors in the task do not suggest any obvious ad hoc ormental strategy (as would be present for example in a case such as 7250 ×"400).ii) For purposes of assessment, the possibilities are 3-digit ×"4-digit.5.Int.2Solve word problems involving three two-digit numbers.i) The given factors are such as to require an efficient/standard algorithm(e.g., 76×48×39). Factors in the task do not suggest any obvious ad hoc ormental strategy (as would be present for example in a case such as 50 × 20×15).16[PLSD]

5.MD.1-1Convert among different-sized standard measurement units within a givenmeasurement system (e.g., convert 5 cm to 0.05 m).5.MD.1-2Solve multi-step, real world problems requiring conversion among differentsizedstandard measurement units within a given measurement system.SampleItems5.NBT.1Students extend their understanding of the base-ten system to the relationshipbetween adjacent places, how numbers compare, and how numbers round fordecimals to thousandths. This standard calls for students to reason about themagnitude of numbers. Students should work with the idea that the tens placeis ten times as much as the ones place, and the ones place is 1/10 th the size ofthe tens place. In fourth grade, students examined the relationships of thedigits in numbers for whole numbers only. This standard extends thisunderstanding to the relationship of decimal fractions. Students use base tenblocks, pictures of base ten blocks, and interactive images of base ten blocksto manipulate and investigate the place value relationships. They use theirunderstanding of unit fractions to compare decimal places and fractionallanguage to describe those comparisons.Before considering the relationship of decimal fractions, students expresstheir understanding that in multi-digit whole numbers, a digit in one placerepresents 10 times what it represents in the place to its right and 1/10 of whatit represents in the place to its left.Example: The 2 in the number 542 is different from the value of the 2 in 324.The 2 in 542 represents 2 ones or 2, while the 2 in 324 represents 2 tens or 20.Since the 2 in 324 is one place to the left of the 2 in 542 the value of the 2 is10 times greater. Meanwhile, the 4 in 542 represents 4 tens or 40 and the 4 in324 represents 4 ones or 4. Since the 4 in 324 is one place to the right of the 4in 542 the value of the 4 in the number 324 is 1/10 th of its value in thenumber 542.Example: A student thinks, “I know that in the number 5555, the 5 in the tensplace (5555) represents 50 and the 5 in the hundreds place (5555) represents500. So a 5 in the hundreds place is ten times as much as a 5 in the tens placeor a 5 in the tens place is 1/10 of the value of a 5 in the hundreds place.17[PLSD]

Base on the base-10 number system digits to the left are times as great asdigits to the right; likewise, digits to the right are 1/10th of digits to the left.For example, the 8 in 845 has a value of 800 which is ten times as much asthe 8 in the number 782. In the same spirit, the 8 in 782 is 1/10th the value ofthe 8 in 845.To extend this understanding of place value to their work with decimals,students use a model of one unit; they cut it into 10 equal pieces, shade in, ordescribe 1/10 of that model using fractional language (“This is 1 out of 10equal parts. So it is 1/10”. I can write this using 1/10 or 0.1”). They repeat theprocess by finding 1/10 of a 1/10 (e.g., dividing 1/10 into 10 equal parts toarrive at 1/100 or 0.01) and can explain their reasoning, “0.01 is 1/10 of 1/10thus is 1/100 of the whole unit.”In the number 55.55, each digit is 5, but the value of the digits is differentbecause of the placement.The 5 that the arrow points to is 1/10 of the 5 to the left and 10 times the 5 tothe right. The 5 in the ones place is 1/10 of 50 and 10 times five tenths.The 5 that the arrow points to is 1/10 of the 5 to the left and 10 times the 5 tothe right. The 5 in the tenths place is 10 times five hundredths.5.NBT.2New at <strong>Grade</strong> 5 is the use of whole number exponents to denote powers of10. Students understand why multiplying by a power of 10 shifts the digits of18[PLSD]

a whole number or decimal that many places to the left.Example: Multiplying by 104 is multiplying by 10 four times. Multiplying by10 once shifts every digit of the multiplicand one place to the left in theproduct (the product is ten times as large) because in the base-ten system thevalue of each place is 10 times the value of the place to its right. Somultiplying by 10 four times shifts every digit 4 places to the left. Patterns inthe number of 0s in products of a whole numbers and a power of 10 and thelocation of the decimal point in products of decimals with powers of 10 canbe explained in terms of place value. Because students have developed theirunderstandings of and computations with decimals in terms of multiplesrather than powers, connecting the terminology of multiples with that ofpowers affords connections between understanding of multiplication andexponentiation. (Progressions for the CCSSM, Number and Operation inBase Ten, CCSS Writing Team, April 2011, page 16)This standard includes multiplying by multiples of 10 and powers of 10,including 10 2 which is 10 x 10=100, and 10 3 which is 10 x 10 x 10=1,000.Students should have experiences working with connecting the pattern of thenumber of zeros in the product when you multiply by powers of 10.Example: 2.5 x 10 3 = 2.5 x (10 x 10 x 10) = 2.5 x 1,000 = 2,500 Studentsshould reason that the exponent above the 10 indicates how many places thedecimal point is moving (not just that the decimal point is moving but thatyou are multiplying or making the number 10 times greater three times) whenyou multiply by a power of 10. Since we are multiplying by a power of 10 thedecimal point moves to the right.350÷10 3 =350÷1,000=0.350=0.35 350/10=35,35/10=3.53.5/10=.0.35,or350x1/10,35x1/10, 3.5 x 1/10 this will relate well tosubsequent work with operating with fractions. This example shows thatwhen we divide by powers of 10, the exponent above the 10 indicates howmany places the decimal point is moving (how many times we are dividingby 10 , the number becomes ten times smaller). Since we are dividing bypowers of 10, the decimal point moves to the left.Students need to be provided with opportunities to explore this concept andcome to this understanding; this should not just be taught procedurally.19[PLSD]

Example: Students might write:36x10=36x10 1 =36036x10x10=36x10 2 =360036x10x10x10=36x10 3 =36,00036x10x10x10x10=36x10 4 =360,000Students might think and/or say:• I noticed that every time, I multiplied by 10 I added a zero to the end of thenumber. That makes sense because each digit’s value became 10 timeslarger. To make a digit 10 times larger, I have to move it one placevalue to the left.• When I multiplied 36 by 10, the 30 became 300. The 6 became 60 or the 36became 360. So I had to add a zero at the end to have the 3 represent 3one-hundreds (instead of 3 tens) and the 6 represents 6 tens (instead of6 ones). Students should be able to use the same type of reasoning asabove to explain why the following multiplication and divisionproblem by powers of 10 make sense.• 523 x 10 3 = 523,000• 5.223 x 10 2 = 522.3• 52.3 ÷ 10 1 = 5.23The place value of 523 is increased by 3 places. The place value of 5.223 isincreased by 2 places. The place value of 52.3 is decreased by one place.5.NBT.3This standard references expanded form of decimals with fractions included.Students should build on their work from Fourth <strong>Grade</strong>, where they workedwith both decimals and fractions interchangeably. Expanded form is includedto build upon work in 5.NBT.2 and deepen students’ understanding of placevalue. Students build on the understanding they developed in fourth grade to20[PLSD]

ead, write, and compare decimals to thousandths. They connect their priorexperiences with using decimal notation for fractions and addition offractions with denominators of 10 and 100. They use concrete models andnumber lines to extend this understanding to decimals to the thousandths.Models may include base ten blocks, place value charts, grids, pictures,drawings, manipulatives, technology-based, etc. They read decimals usingfractional language and write decimals in fractional form, as well as inexpanded notation. This investigation leads them to understandingequivalence of decimals (0.8 = 0.80 = 0.800).Comparing decimals builds on work from fourth grade.Example: Some equivalent forms of 0.72 are:72/100 7/10 + 2/100 7 x (1/10) + 2 x (1/100) 0.70 + 0.0270/100 + 2/100 0.720 7 x (1/10) + 2 x (1/100) + 0 x (1/1000) 720/1000Students need to understand the size of decimal numbers and relate them tocommon benchmarks such as 0, 0.5 (0.50 and 0.500), and 1. Comparingtenths to tenths, hundredths to hundredths, and thousandths to thousandths issimplified if students use their understanding of fractions to comparedecimals.Example: Comparing 0.25 and 0.17, a student might think, “25 hundredths ismore than 17 hundredths”. They may also think that it is 8 hundredths more.They may write this comparison as 0.25 > 0.17 and recognize that 0.17 < 0.25is another way to express this comparison.Comparing 0.207 to 0.26, a student might think, “Both numbers have 2tenths, so I need to compare the hundredths. The second number has 6hundredths and the first number has no hundredths so the second numbermust be larger. Another student might think while writing fractions, “I knowthat 0.207 is 207 thousandths (and may write 207/1000). 0.26 is 26hundredths (and may write 26/100) but I can also think of it as 260thousandths (260/1000). So, 260 thousandths is more than 207 thousandths.5.NBT.4This standard refers to rounding. Students should go beyond simply applyingan algorithm or procedure for rounding. The expectation is that students havea deep understanding of place value and number sense and can explain andreason about the answers they get when they round. Students should have21[PLSD]

numerous experiences using a number line to support their work withrounding.Example: Round 14.235 to the nearest tenth. Students recognize that thepossible answer must be in tenths thus, it is either 14.2 or 14.3. They thenidentify that 14.235 is closer to 14.2 (14.20) than to 14.3 (14.30).Students should use benchmark numbers to support this work. Benchmarksare convenient numbers for comparing and rounding numbers. 0., 0.5, 1, 1.5are examples of benchmark numbers.Example: Which benchmark number is the best estimate of the shadedamount in the model below? Explain your thinking.5.NBT.5In fifth grade, students fluently compute products of whole numbers using thestandard algorithm. Underlying this algorithm are the properties of operationsand the base-ten system. Division strategies in fifth grade involve breakingthe dividend apart into like base-ten units and applying the distributiveproperty to find the quotient place by place, starting from the highest place.(Division can also be viewed as finding an unknown factor: the dividend isthe product, the divisor is the known factor, and the quotient is the unknownfactor.) Students continue their fourth grade work on division, extending it tocomputation of whole number quotients with dividends of up to four digitsand two-digit divisors. Estimation becomes relevant when extending to twodigitdivisors. Even if students round appropriately, the resulting estimatemay need to be adjusted.22[PLSD]

Computation algorithm. A set of predefined steps applicable to a class ofproblems that gives the correct result in every case when the steps are carriedout correctly. Computation strategy. Purposeful manipulations that may bechosen for specific problems, may not have a fixed order, and may be aimedat converting one problem into another.This standard refers to fluency which means accuracy (correct answer),efficiency (a reasonable amount of steps), and flexibility (using strategiessuch as the distributive property or breaking numbers apart also usingstrategies according to the numbers in the problem, 26 x 4 may lend itself to(25 x 4 ) + 4 where as another problem might lend itself to making anequivalent problem 32 x 4 = 64 x 2)). This standard builds upon students’work with multiplying numbers in third and fourth grade. In fourth grade,students developed understanding of multiplication through using variousstrategies. While the standard algorithm is mentioned, alternative strategiesare also appropriate to help students develop conceptual understanding. Thesize of the numbers should NOT exceed a three-digit factor by a two-digitfactor.Examples of alternative strategies: There are 225 dozen cookies in the bakery.How many cookies are there?Draw a array model for 225 x 12.... 200 x 10, 200 x 2, 20 x 10, 20 x 2, 5 x 10,23[PLSD]

5 x 2 225 x 125.NBT.6This standard references various strategies for division. Division problemscan include remainders. Even though this standard leads more towardscomputation, the connection to story contexts is critical. Make sure studentsare exposed to problems where the divisor is the number of groups and wherethe divisor is the size of the groups. In fourth grade, students’ experienceswith division were limited to dividing by one-digit divisors. This standardextends students’ prior experiences with strategies, illustrations, andexplanations. When the two-digit divisor is a “familiar” number, a studentmight decompose the dividend using place value.Example: There are 1,716 students participating in Field Day. They are putinto teams of 16 for the competition. How many teams get created? If youhave left over students, what do you do with them?24[PLSD]

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5.NBT.7Because of the uniformity of the structure of the base-ten system, students usethe same place value understanding for adding and subtracting decimals thatthey used for adding and subtracting whole numbers. Like base-ten units mustbe added and subtracted, so students need to attend to aligning thecorresponding places correctly (this also aligns the decimal points). It canhelp to put 0s in places so that all numbers show the same number of placesto the right of the decimal point. Although whole numbers are not usuallywritten with a decimal point, but that a decimal point with 0s on its right canbe inserted (e.g., 16 can also be written as 16.0 or 16.00). The process ofcomposing and decomposing a base-ten unit is the same for decimals as forwhole numbers and the same methods of recording numerical work can beused with decimals as with whole numbers. For example, students can writedigits representing new units below on the addition or subtraction line, and26[PLSD]

they can decompose units wherever needed before subtracting.General methods used for computing products of whole numbers extend toproducts of decimals. Because the expectations for decimals are limited tothousandths and expectations for factors are limited to hundredths at thisgrade level, students will multiply tenths with tenths and tenths withhundredths, but they need not multiply hundredths with hundredths. Beforestudents consider decimal multiplication more generally, they can study theeffect of multiplying by 0.1 and by 0.01 to explain why the product is ten or ahundred times as small as the multiplicand (moves one or two places to theright). They can then extend their reasoning to multipliers that are single-digitmultiples of 0.1 and 0.01 (e.g., 0.2 and 0.02, etc.).There are several lines of reasoning that students can use to"explain theplacement of the decimal point in other products of decimals. Students canthink about the product of the smallest base-ten units of each factor. Forexample, a tenth times a tenth is a hundredth, so 3.2 x 7.1 will have an entryin the hundredth place. Note, however, that students might place the decimalpoint incorrectly for 3.2 x 8.5 unless they take into account the 0 in the onesplace of 32 x 85. (Or they can think of 0.2 x 0.5 as 10 hundredths.) Studentscan also think of the decimals as fractions or as whole numbers divided by 10or 100. 5.NF.3 When they place the decimal point in the product, they have todivide by a 10 from each factor or 100 from one factor. For example, to seethat 0.6 x 0.8 = 0.48, students can use fractions: 6/10 x 8/10 = 48/100. 5.NF.4Students can also reason that when they carry out the multiplication withoutthe decimal point, they have multiplied each decimal factor by 10 or 100, sothey will need to divide by those numbers in the end to get the correct answer.Also, students can use reasoning about the sizes of numbers to determine theplacement of the decimal point. For example, 3.2 x 8.5 should be close to 3 x9, so 27.2 is a more reasonable product for 3.2 x 8.5 than 2.72 or 272. Thisestimation-based method is not reliable in all cases, however, especially incases students will encounter in later grades. For example, it is not easy todecide where to place the decimal point in 0.023 x 0.0045 based onestimation. Students can summarize the results of their reasoning such asthose above as specific numerical patterns and then as one general overallpattern such as “the number of decimal places in the product is the sum of thenumber of decimal places in each factor.” General methods used forcomputing quotients of whole numbers extend to decimals with the additionalissue of placing the decimal point in the quotient. As with decimalmultiplication, students can first examine the cases of dividing by 0.1 and27[PLSD]

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Additional multiplication and division examples:5.MD.1Calls for students to convert measurements within the same system ofmeasurement in the context of multi-step, real-world problems. Bothcustomary and standard measurement systems are included; students workedwith both metric and customary units of length in second grade. In thirdgrade, students work with metric units of mass and liquid volume. In fourth31[PLSD]

grade, students work with both systems and begin conversions within systemsin length, mass and volume.Students should explore how the base-ten system supports conversions withinthe metric system. Example: 100 cm = 1 meter.In <strong>Grade</strong> 5, students extend their abilities from <strong>Grade</strong> 4 to expressmeasurements in larger or smaller units within a measurement system. This isan excellent opportunity to reinforce notions of place value for wholenumbers and decimals, and connection between fractions and decimals (e.g.,2 1⁄2 meters can be expressed as 2.5 meters or 250 centimeters). For example,building on the table from <strong>Grade</strong> 4, <strong>Grade</strong> 5 students might complete a tableof equivalent measurements in feet and inches. <strong>Grade</strong> 5 students also learnand use such conversions in solving multi-step, real world problems (seeexample below).32[PLSD]

I CanStatements5.NBT.1• In a multi-digit number, the value of a digit in one particular place isten times the value of the same digit in the place to its right.• Similarly, the value of a digit in one particular place is one-tenth of thevalue of the same digit in the place to its left.5.NBT.2• To multiply a whole number by a power of 10 that is greater than 1,add on as many 0's as appear in the power of ten.• To divide by a power of ten, start from the right of the whole numberand move the decimal to the left the same number of places as there are0's in the power of ten.5.NBT.3• I can read decimals to the thousandths place.• I can write decimals to the thousandths place.• I can compare two decimals to the thousandths place using >,

5.NBT.4• I can use place value to round decimals.5.NBT.5• I can multiply multi-digit whole numbers.5.NBT.6• I can divide four digit dividends by two digit divisors.• I can use place value strategies to divide.• I can use the inverse operation of multiplication and division.• I can find quotients by using an equation.• I can find quotients by using rectangular arrays and/or area models.5.NBT.7• I can add decimals to the hundredths.• I can subtract decimals to hundredths.• I can multiply decimals to hundredths.• I can divide decimals to hundredths.• I can relate the strategy used to a written method and explain thereasoning used.• I can demonstrate computations by using models and drawings.5.MD.1• I can convert measurements within the same measuring system.Vocabulary• rounding• sequence• standard form• subtrahend• sum• tenth• tenths• term• thousandth• thousandths• addend• Associative Property of Addition• base-ten numeral form• benchmark• Commutative Property of Addition34[PLSD]

• compose• decimal• decimal point• decompose• difference• estimate• expanded form• greater than• hundredth• hundredths• inequality• less than• minuend• place value• capacity• centimeter• cup• customary system• decimeter• dekameter• elapsed time• fluid ounce• foot• gallon• gram• inch• kilogram• kilometer• liter• mass• meter• metric system• mile• milligram• milliliter• millimeter• ounce35[PLSD]

InstructionalStrategies• pint• pound• quart• ton• weight• yard5.NBT.1-4Instructional StrategiesIn <strong>Grade</strong> 5, the concept of place value is extended to include decimal valuesto thousandths. The strategies for <strong>Grade</strong>s 3 and 4 should be drawn upon andextended for whole numbers and decimal numbers. For example, studentsneed to continue to represent, write and state the value of numbers includingdecimal numbers. For students who are not able to read, write and representmulti-digit numbers, working with decimals will be challenging.Money is a good medium to compare decimals. Present contextual situationsthat require the comparison of the cost of two items to determine the lower orhigher priced item. Students should also be able to identify how manypennies, dimes, dollars and ten dollars, etc., are in a given value. Helpstudents make connections between the number of each type of coin and thevalue of each coin, and the expanded form of the number. Build on theunderstanding that it always takes ten of the number to the right to make thenumber to the left.Number cards, number cubes, spinners and other manipulatives can be usedto generate decimal numbers. For example, have students roll three numbercubes, then create the largest and small number to the thousandths place. Askstudents to represent the number with numerals and words.Instructional Resources/ToolsNation Library of Virtual Manipulatives; Base Block Decimals, Student use aTen Frames to demonstrate decimal relationships.http://nlvm.usu.edu/en/nav/frames_asid_264_g_3_t_1.html?from=search.html?qt=decimalCommon MisconceptionsA common misconception that students have when trying to extend theirunderstanding of whole number place value to decimal place value is that asyou move to the left of the decimal point, the number increases in value.36[PLSD]

Reinforcing the concept of powers of ten is essential for addressing this issue.A second misconception that is directly related to comparing whole numbersis the idea that the longer the number the greater the number. With wholenumbers, a 5-digit number is always greater that a 1-, 2-, 3-, or 4-digitnumber. However, with decimals a number with one decimal place may begreater than a number with two or three decimal places. For example, 0.5 isgreater than 0.12, 0.009 or 0.499. One method for comparing decimals it tomake all numbers have the same number of digits to the right of the decimalpoint by adding zeros to the number, such as 0.500, 0.120, 0.009 and 0.499. Asecond method is to use a place-value chart to place the numerals forcomparison.5.NBT.5-7Because students have used various models and strategies to solve problemsinvolving multiplication with whole numbers, they should be able totransition to using standard algorithms effectively. With guidance from theteacher, they should understand the connection between the standardalgorithm and their strategies.Connections between the algorithm for multiplying multi-digit wholenumbers and strategies such as partial products or lattice multiplication arenecessary for students’ understanding.You can multiply by listing all the partial products. For example:The multiplication can also be done without listing the partial products bymultiplying the value of each digit from one factor by the value of each digitfrom the other factor. Understanding of place value is vital in using thestandard algorithm.In using the standard algorithm for multiplication, when multiplying the ones,32 ones is 3 tens and 2 ones. The 2 is written in the ones place. Whenmultiplying the tens, the 24 tens is 2 hundreds and 4 tens. But, the 3 tens fromthe 32 ones need to be added to these 4 tens, for 7 tens. Multiplying the37[PLSD]

hundreds, the 16 hundreds is 1 thousand and 6 hundreds. But, the 2 hundredsfrom the 24 tens need to be added to these 6 hundreds, for 8 hundreds.As students developed efficient strategies to do whole number operations,they should also develop efficient strategies with decimal operations.Students should learn to estimate decimal computations before they computewith pencil and paper. The focus on estimation should be on the meaning ofthe numbers and the operations, not on how many decimal places areinvolved. For example, to estimate the product of 32.84 × 4.6, the estimatewould be more than 120, closer to 150. Students should consider that 32.84 iscloser to 30 and 4.6 is closer to 5. The product of 30 and 5 is 150. Therefore,the product of 32.84 × 4.6 should be close to 150.Have students use estimation to find the product by using exactly the samedigits in one of the factors with the decimal point in a different position eachtime. For example, have students estimate the product of 275 × 3.8; 27.5 ×3.8 and 2.75 × 3.8, and discuss why the estimates should or should not be thesame.Instructional Resources/ToolsDecimal place-value chartFrom the National Library of Virtual Manipulatives: Base Blocks Decimals–Add and subtract decimal valuesusing base blocks. (Note: make sure the Base equals 10).Common MisconceptionsStudents might compute the sum or difference of decimals by lining up theright-hand digits as they would whole number. For example, in computing thesum of 15.34 + 12.9, students will write the problem in this manner:To help students add and subtract decimals correctly, have them first estimatethe sum or difference. Providing students with a decimal-place value chart38[PLSD]

will enable them to place the digits in the proper place.5.MD.1Instructional StrategiesStudents should gain ease in converting units of measures in equivalent formswithin the same system. To convert from one unit to another unit, therelationship between the units must be known. In order for students to have abetter understanding of the relationships between units, they need to usemeasuring tools in class. The number of units must relate to the size of theunit. For example, students have discovered that there are 12 inches in 1 footand 3 feet in 1 yard. This understanding is needed to convert inches to yards.Using 12-inch rulers and yardsticks, students can see that three of the 12-inchrulers are equivalent to one yardstick (3 × 12 inches = 36 inches; 36 inches =1 yard). Using this knowledge, students can decide whether to multiply ordivide when making conversions.Once students have an understanding of the relationships between units andhow to do conversions, they are ready to solve multi-step problems thatrequire conversions within the same system. Allow students to discussmethods used in solving the problems. Begin with problems that allow forrenaming the units to represent the solution before using problems thatrequire renaming to find the solution.Instructional Resources/ToolsYardsticks(meter sticks) and rulers (marked with customary and metric units)Teaspoons and tablespoons Graduated measuring cups (marked withcustomary and metric units) From the National Council of Teachers of<strong>Math</strong>ematics, Illuminations: - Discovering Gallon Man.http://illuminations.nctm.org/LessonDetail.aspx?ID=L513Students experiment with units of liquid measure used in the customarysystem of measurement. They practice making volume conversions in thecustomary system.From the National Council of Teachers of <strong>Math</strong>ematics, Illuminations: – DoYou Measure Up?http://illuminations.nctm.org/LessonDetail.aspx?ID=L512Students learn the basics of the metric system. They identify which units ofmeasurement are used to measure specific objects, and they learn to convert39[PLSD]

etween units within the same system.Common MisconceptionsWhen solving problems that require renaming units, students use theirknowledge of renaming the numbers as with whole numbers. Students need topay attention to the unit of measurement which dictates the renaming and thenumber to use. The same procedures used in renaming whole numbers shouldnot be taught when solving problems involving measurement conversions.For example, when subtracting 5 inches from 2 feet, students may take onefoot from the 2 feet and use it as 10 inches. Since there were no inches withthe 2 feet, they put 1 with 0 inches and make it 10 inches.Connections This cluster is connected to the <strong>Grade</strong> 5 Critical Area of Focus #2, Extendingdivision to 2-digit divisors, integrating decimal fractions into the place valuesystem and developing understanding of operations with decimals tohundredths, and developing fluency with whole number and decimaloperations.CRITICAL AREA OF FOCUS #2Extending division to 2-digit divisors, integrating decimal fractions into theplace value system and developing understanding of operations with decimalsto hundredths, and developing fluency with whole number and decimaloperationsStudents develop understanding of why division procedures work based onthe meaning of base-ten numerals and properties of operations. They finalizefluency with multi-digit addition, subtraction, multiplication, and division.They apply their understandings of models for decimals, decimal notation,and properties of operations to add and subtract decimals to hundredths. Theydevelop fluency in these computations, and make reasonable estimates oftheir results. Students use the relationship between decimals and fractions, aswell as the relationship between finite decimals and whole numbers (i.e., afinite decimal multiplied by an appropriate power of 10 is a whole number),to understand and explain why the procedures for multiplying and dividingfinite decimals make sense. They compute products and quotients of decimals40[PLSD]

to hundredths efficiently and accurately.ResourcesPlace Value (include Powers of Ten)Cosmic Voyage Clip - narrated by Morgan Freeman -http://www.youtube.com/watch?v=qxXf7AJZ73APowers of 10 - Charles and Ray Eames (original movie clip) -http://www.youtube.com/watch?v=38ti9BJiyvsLearnAlberta - Place Value - Video Tutorial -http://www.learnalberta.ca/content/me5l/html/<strong>Math</strong>5.html?launch=trueEducation Place - Place Value - Student Tutorial –http://eduplace.com/cgibin/schtemplate.cgi?template=/math/hmm/models/tm_popup.thtml&grade=5&chapter=1&lesson=1&title=Place+Value+Through+Hundred+Thousands&tm=tmff0101eMr. Nussbaum - Decimals of the Caribbean - Game -http://www.mrnussbaum.com/docrb1.htmMr. Nussbaum - Place Value Pirates - Game -http://www.mrnussbaum.com/placevaluepirates.htmThe Scale of the Universe - Powers of Ten - Demonstration Model -http://htwins.net/scale2/scale2.swf?bordercolor=whiteComparing DecimalsUEN - Patterns with Decimals - Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=6165Learn Alberta - Comparing and Ordering Decimals - Video Tutorial -http://www.learnalberta.ca/content/me5l/html/<strong>Math</strong>5.html?launch=trueBBC - Builder Ted - Game -http://www.bbc.co.uk/education/mathsfile/shockwave/games/laddergame.htmlDecimal Squares - Rope Tug - Game -http://www.decimalsquares.com/dsGames/games/tugowar.html41[PLSD]

Rounding DecimalsBBC - Rounding Off - Game -http://www.bbc.co.uk/education/mathsfile/shockwave/games/roundoff.htmlDecimal Squares - Laser Beams - Game -http://decimalsquares.com/dsGames/games/laserbeam.htmlScholastic Study Jams - Rounding Decimals - Student Tutorial -http://studyjams.scholastic.com/studyjams/jams/math/decimalspercents/rounding-decimals.htmMr. Nussbaum - Half-court rounding - Game -http://www.mrnussbaum.com/rounding/index.htmlMr. Nussbaum - Rounding Master - Game -http://www.mrnussbaum.com/mathmillions/index.htmlCustomary/Standard SystemEasy Surf - Converter Applet –http://www.easysurf.cc/cnver13.htm#ctog1BBC - Animal Weigh In - Game -http://www.bbc.co.uk/education/mathsfile/shockwave/games/animal.htmlThe Teacher Website - Gallon Man - Lesson -http://www.theteacherwebsite.com/mrgallonmanproject-tools.pdfHMH School Publishers - Game -http://www.harcourtschool.com/activity/con_math/g04c24.htmlMetric SystemAtlantis Ed. - Teacher Tutorial -http://atlantis.coe.uh.edu/archive/science/science_lessons/scienceles3/metric/metric.htmlUEN - Lesson –http://www.uen.org/Lessonplan/preview.cgi?LPid=21571Purple <strong>Math</strong> - Teacher Tutorial -http://www.purplemath.com/modules/metric.htm42[PLSD]

Figure This - Problem Solving with Measurement -http://www.figurethis.org/challenges/c67/challenge.htm<strong>Math</strong> Playground - Student Tutorial Video -http://www.mathplayground.com/howto_Metric.html43[PLSD]

Unit 2: 5 th <strong>Grade</strong> Quarters1 and 2Approximately3 WeeksDomain: Operations and Algebraic Thinking and GeometryStandards5.OA.1. Use parentheses, brackets, or braces in numerical expressions, andevaluate expressions with these symbols.5.OA.2. Write simple expressions that record calculations with numbers, andinterpret numerical expressions without evaluating them.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.5OA.3. Generate two numerical patterns using two given rules. Identifyapparent relationships between corresponding terms. Form ordered pairsconsisting of corresponding terms from the two patterns, and graph theordered pairs on a coordinate plane.For example, given the rule “Add 3” and the starting number 0, and giventhe rule “Add 6” and the starting number 0, generate terms in the resultingsequences, and observe that the terms in one sequence are twice thecorresponding terms in the other sequence. Explain informally why this is so.5.G.1. Use a pair of perpendicular number lines, called axes, to define acoordinate system, with the intersection of the lines (the origin) arranged tocoincide with the 0 on each line and a given point in the plane located byusing an ordered pair of numbers, called its coordinates. Understand that thefirst number indicates how far to travel from the origin in the direction of oneaxis, and the second number indicates how far to travel in the direction of thesecond axis, with the convention that the names of the two axes and thecoordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).5.G.2. Represent real world and mathematical problems by graphing points inthe first quadrant of the coordinate plane, and interpret coordinate values ofpoints in the context of the situation.44[PLSD]

PARCCEOYClarification5.OA.1Use parentheses, brackets, or braces in numerical expressions, and evaluateexpressions with these symbols.5.OA.2-1Write simple expressions that record calculations with numbers. For example,express the calculation “add 8 and 7,thenmultiplyby2”as 2×(8+7).i) Note that expressions elsewhere in CCSS are thought of as recordingcalculations with numbers (or letters standing for numbers) as well; see forexample 6.EE.2a. See also the first paragraph of the Progression forExpressions and Equations.5.OA.2-2Interpret numerical expressions without evaluating them. For example,recognize that 3× (18932 + 921) is three times as large as 18932 + 921without having to calculate the indicated sum or product.5.OA.3Generate two numerical patterns using two given rules. Identify apparentrelationships between corresponding terms. Form ordered pairs consisting ofcorresponding terms from the two patterns, and graph the ordered pairs on acoordinate plane. For example, given the rule “Add 3” and the startingnumber 0, and given the rule “Add 6” and the starting number 0, generateterms in the resulting sequences, and observe that the terms in one sequenceare twice the corresponding terms in the other sequence. Explain informallywhy this is so.5.G.1Use a pair of perpendicular number lines, called axes, to define a coordinatesystem, with the intersection of the lines (the origin) arranged to coincidewith the 0 on each line and a given point in the plane located by using anordered pair of numbers, called its coordinates. Understand that the firstnumber indicates how far to travel from the origin in the direction of one axis,and the second number indicates how far to travel in the direction of thesecond axis, with the convention that the names of the two axes and the45[PLSD]

coordinates correspond (e.g., x"axis&and x"coordinate,&y"axis&and y"coordinate).&i) Tasks probe student understanding of the coordinate plane as arepresentation scheme, with essential features as articulated in standard 5.G.1.ii) It is appropriate for tasks involving only plotting of points to be aligned tothis evidence statement.5.G.2Represent real world and mathematical problems by graphing points in thefirst quadrant of the coordinate plane, and interpret coordinate values ofpoints in the context of the situation.SampleItems5.OA.1Example:To further develop students’ understanding of grouping symbols and facilitywith operations, students place grouping symbols in equations to make theequations true or they compare expressions that are grouped differently.5.OA.246[PLSD]

5.OA.3This standard extends the work from Fourth <strong>Grade</strong>, where students generatenumerical patterns when they are given one rule. In Fifth <strong>Grade</strong>, students aregiven two rules and generate two numerical patterns. The graphs that arecreated should be line graphs to represent the pattern. This is a linear functionwhich is why we get the straight lines. The Days are the independent variable,Fish are the dependent variables, and the constant rate is what the ruleidentifies in the table.Example: Describe the pattern: Since Terri catches 4 fish each day, and Samcatches 2 fish, the amount of Terri’s fish is always greater. Terri’s fish is alsoalways twice as much as Sam’s fish. Today, both Sam and Terri have no fish.They both go fishing each day. Sam catches 2 fish each day. Terri catches 4fish each day. How many fish do they have after each of the five days? Makea graph of the number of fish.Plot the points on a coordinate plane and make a line graph, and then interpretthe graph. Student: My graph shows that Terri always has more fish than47[PLSD]

Sam. Terri’s fish increases at a higher rate since she catches 4 fish every day.Sam only catches 2 fish every day, so his number of fish increases at asmaller rate than Terri. Important to note as well that the lines becomeincreasingly further apart. Identify apparent relationships betweencorresponding terms. Additional relationships: The two lines will neverintersect; there will not be a day in which boys have the same total of fish,explain the relationship between the number of days that has passed and thenumber of fish a boy has (2n or 4n, n being the number of days).Example: Use the rule “add 3” to write a sequence of numbers. Starting witha 0, students write 0, 3, 6, 9, 12, . . . Use the rule “add 6” to write a sequenceof numbers. Starting with 0, students write 0, 6, 12, 18, 24, . . .After comparing these two sequences, the students notice that each term inthe second sequence is twice the corresponding terms of the first sequence.One way they justify this is by describing the patterns of the terms. Theirjustification may include some mathematical notation (See example below).A student may explain that both sequences start with zero and to generateeach term of the second sequence he/she added 6, which is twice as much aswas added to produce the terms in the first sequence. Students may also usethe distributive property to describe the relationship between the twonumerical patterns by reasoning that 6 + 6 + 6 = 2 (3 + 3 + 3).0, +3 3, +3 6, +3 9, +3 12,...0, +6 6, +6 12, +6 18, +6 24, ...Once students can describe that the second sequence of numbers is twice the48[PLSD]

corresponding terms of the first sequence, the terms can be written in orderedpairs and then graphed on a coordinate grid. They should recognize that eachpoint on the graph represents two quantities in which the second quantity istwice the first quantity.5.G.1 and 5.G.2 These standards deal with only the first quadrant (positivenumbers) in the coordinate plane. Although students can often “locate apoint,” these understandings are beyond simple skills. For example, initially,students often fail to distinguish between two different ways of viewing thepoint (2, 3), say, as instructions: “right 2, up 3”; and as the point defined bybeing a distance 2 from the y-axis and a distance 3 from the x-axis. In thesetwo descriptions the 2 is first associated with the x-axis, then with the y-axis.Example:Plot these points on a coordinate grid.Point A: (2,6) Point B: (4,6) Point C: (6,3) Point D: (2,3)49[PLSD]

Connect the points in order. Make sure to connect Point D back to Point A.1. What geometric figure is formed? What attributes did you use toidentify it?2. What line segments in this figure are parallel?3. What line segments in this figure are perpendicular?solutions: trapezoid, line segments AB and DC are parallel, segments AD andDC are perpendicularExample:Emanuel draws a line segment from (1, 3) to (8, 10). He then draws a linesegment from (0, 2) to (7, 9). If he wants to draw another line segment that isparallel to those two segments what points will he use?This standard references real-world and mathematical problems, including thetraveling from one point to another and identifying the coordinates of missingpoints in geometric figures, such as squares, rectangles, and parallelograms.Example:Sara has saved \$20. She earns \$8 for each hour she works.If Sara saves all of her money, how much will she have after working 3hours? 5 hours? 10 hours?Create a graph that shows the relationship between the hours Sara worked andthe amount of money she has saved. What other information do you knowfrom analyzing the graph?Example:50[PLSD]

Use the graph below to determine how much money Jack makes afterworking exactly 9 hours.I CanStatements5.OA.1• Evaluate/solve simple expressions which include brackets, braces,parentheses• Write simple expressions without solving5.OA.2• Write expressions that record calculations with numbers• Interpret numerical expressions without evaluating them5.OA.3• Generate a patterns from a numerical rules• Identify the relationships between the corresponding terms of two rules• Represent those patterns in a function table• Graph ordered pairs in quadrant one (+,+)5.G.1• I can understand how to graph ordered pairs on a coordinate plane5.G.2• I can graph and interpret points in the first quadrant of a coordinateplane.Vocabulary• Additive Identity Property of 0• algorithm• area model51[PLSD]

• array• Associative Property of Addition• Associative Property of Multiplication• base of an exponent• braces• brackets• Commutative Property of Addition• Commutative Property of Multiplication• Distributive Property• dividend• divisor• equation• estimate• evaluate• exponent• expression• factor• inverse operations• long division• Multiplicative Identity Property of 1• multiply• numerical expression• Order of Operations• parentheses• period• place value• powers of ten• product• quotient• remainder• sum• whole numbers• line plot• ordered pair• origin• perpendicular• plane52[PLSD]

InstructionalStrategies• quadrants• scale• sequence• unit fraction• x-axis• x-coordinate• y-axis• y-coordinate• axis (axes)• coordinate plane• coordinate system• coordinates• corresponding terms• data• fraction• intersect• interval• line graph5.OA.1 and 5.OA.2Instructional StrategiesStudents should be given ample opportunities to explore numericalexpressions with mixed operations. This is the foundation for evaluatingnumerical and algebraic expressions that will include whole-numberexponents in <strong>Grade</strong> 6.There are conventions (rules) determined by mathematicians that must belearned with no conceptual basis. For example, multiplication and divisionare always done before addition and subtraction.Begin with expressions that have two operations without any groupingsymbols (multiplication or division combined with addition or subtraction)before introducing expressions with multiple operations. Using the samedigits, with the operations in a different order, have students evaluate theexpressions and discuss why the value of the expression is different. Forexample, have students evaluate 5 × 3 + 6 and 5 + 3 × 6. Discuss the rulesthat must be followed. Have students insert parentheses around themultiplication or division part in an expression. A discussion should focus onthe similarities and differences in the problems and the results. This leads tostudents being able to solve problem situations which require that they know53[PLSD]

the order in which operations should take place.After students have evaluated expressions without grouping symbols, presentproblems with one grouping symbol, beginning with parentheses, then incombination with brackets and/or braces.Have students write numerical expressions in words without calculating thevalue. This is the foundation for writing algebraic expressions. Then, havestudents write numerical expressions from phrases without calculating them.Instructional Resources/ToolsCalculators (scientific or four-function)The Ohio Resource Center ORC # 11463 From the National Council ofTeachers of <strong>Math</strong>ematics, Illuminations: Order of Operations Bingo.http://illuminations.nctm.org/LessonDetail.aspx?id=L730Instead of calling numbers to play Bingo, you call (and write) numericalexpressions to be evaluated for the numbers on the Bingo cards. Theoperations in this lesson are addition, subtraction, multiplication, anddivision; the numbers are all single-digit whole numbers.5.OA.3Instructional StrategiesStudents should have experienced generating and analyzing numericalpatterns using a given rule in <strong>Grade</strong> 4.Given two rules with an apparent relationship, students should be able toidentify the relationship between the resulting sequences of the terms in onesequence to the corresponding terms in the other sequence. For example,starting with 0, multiply by 4 and starting with 0, multiply by 8 and generateeach sequence of numbers (0, 4, 8, 12, 16, ...) and (0, 8, 16, 24, 32,...).Students should see that the terms in the second sequence are double theterms in the first sequence, or that the terms in the first sequence are half theterms in the second sequence.Have students form ordered pairs and graph them on a coordinate plane.Patterns can be also discerned in graphs.Graphing ordered pairs on a coordinate plane is introduced to students in theGeometry domain where students solve real-world and mathematical54[PLSD]

problems. For the purpose of this cluster, only use the first quadrant of thecoordinate plane, which contains positive numbers only. Provide coordinategrids for the students, but also have them make coordinate grids. In <strong>Grade</strong> 6,students will position pairs of integers on a coordinate plane.The graph of both sequences of numbers is a visual representation that willshow the relationship between the two sequences of numbers.Encourage students to represent the sequences in T-charts so that they can seea connection between the graph and the sequences.Instructional Resources/ToolsGrid paper55[PLSD]

Common MisconceptionsStudents reverse the points when plotting them on a coordinate plane. Theycount up first on the y-axis and then count over on the x-axis. The location ofevery point in the plane has a specific place. Have students plot points wherethe numbers are reversed such as (4, 5) and (5, 4). Begin with studentsproviding a verbal description of how to plot each point. Then, have themfollow the verbal description and plot each point.5.G.1 and 5.G.2Instructional StrategiesStudents need to understand the underlying structure of the coordinate systemand see how axes make it possible to locate points anywhere on a coordinateplane. This is the first time students are working with coordinate planes, andonly in the first quadrant. It is important that students create the coordinategrid themselves. This can be related to two number lines and reliance onprevious experiences with moving along a number line.Multiple experiences with plotting points are needed. Provide points plottedon a grid and have students name and write the ordered pair. Have studentsdescribe how to get to the location. Encourage students to articulate directionsas they plot points.Present real-world and mathematical problems and have students graph pointsin the first quadrant of the coordinate plane. Gathering and graphing data is avaluable experience for students. It helps them to develop an understanding ofcoordinates and what the overall graph represents. Students also need toanalyze the graph by interpreting the coordinate values in the context of thesituation.Instructional Resources/ToolsGrid/graph paperFrom the National Council of Teachers of <strong>Math</strong>ematics, Illuminations:Finding Your Way Around - Students explore twodimensionalspace via an activity in which they navigate the coordinate plane.From the National Council of Teachers of <strong>Math</strong>ematics, Illuminations:http://illuminations.nctm.org/LessonDetail.aspx?ID=L28056[PLSD]

Describe the Way – In this lesson, students will review plotting points andlabeling axes. Students generate a set of random points all located in the firstquadrant.http://illuminations.nctm.org/LessonDetail.aspx?id=L777Common MisconceptionsWhen playing games with coordinates or looking at maps, students may thinkthe order in plotting a coordinate point is not important. Have students plotpoints so that the position of the coordinates is switched. For example, havestudents plot (3, 4) and (4, 3) and discuss the order used to plot the points.Have students create directions for others to follow so that they becomeaware of the importance of direction and distance.ConnectionsThis cluster is connected to the <strong>Grade</strong> 5 Critical Area of Focus #2, Extendingdivision to 2-digit divisors, integrating decimal fractions into the place valuesystem and developing understanding of operations with decimals tohundredths, and developing fluency with whole number and decimaloperations.CRITICAL AREA OF FOCUS #2Extending division to 2-digit divisors, integrating decimal fractions into theplace value system and developing understanding of operations with decimalsto hundredths, and developing fluency with whole number and decimaloperationsStudents develop understanding of why division procedures work based onthe meaning of base-ten numerals and properties of operations. They finalizefluency with multi-digit addition, subtraction, multiplication, and division.They apply their understandings of models for decimals, decimal notation,and properties of operations to add and subtract decimals to hundredths. Theydevelop fluency in these computations, and make reasonable estimates oftheir results. Students use the relationship between decimals and fractions, aswell as the relationship between finite decimals and whole numbers (i.e., afinite decimal multiplied by an appropriate power of 10 is a whole number),to understand and explain why the procedures for multiplying and dividingfinite decimals make sense. They compute products and quotients of decimalsto hundredths efficiently and accurately.57[PLSD]

ResourcesOrder of OperationsLearnAlberta - Exploring Order of Operations - Student Interactivehttp://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.INTE&lesson=html/object_interactives/order_of_operations/use_it.htmlIlluminations - Order of Operations Bingo - Lesson -http://illuminations.nctm.org/LessonDetail.aspx?id=L730<strong>Math</strong> Goodies - Order of Operations - Tutorial and Practice Exercises -http://www.mathgoodies.com/lessons/vol7/order_operations.htmlIlluminations - Everything Balances Out in the End - Lesson -http://illuminations.nctm.org/LessonDetail.aspx?ID=L643Illuminations - Exploring Krypto - Lesson -http://illuminations.nctm.org/LessonDetail.aspx?ID=L803Purple <strong>Math</strong> - Order of Operations- Teacher Tutorial -http://www.purplemath.com/modules/orderops2.htm<strong>Math</strong> Playground - Order of Operations - Game -http://www.mathplayground.com/order_of_operations.htmlKahn Academy - Order of Operations - Teacher Tutorial -http://www.khanacademy.org/video/order-of-operations?topic=order-ofoperationsShodor - Order of Operations - Assessment -http://www.shodor.org/interactivate/activities/OperationsQuiz/Shodor - Order of Operations Four - Game -http://www.shodor.org/interactivate/activities/OrderOfOperationsFou/Jefferson Lab - Speed <strong>Math</strong> - Game -http://education.jlab.org/smdeluxe/index.htmlIXL - Simplify Expressions Using Order of Operations - Assessment -http://www.ixl.com/math/grade-5/simplify-expressions-using-order-ofoperations-and-parentheses58[PLSD]

Mr. Nussbaum - The Order of Operations Royal Rescue - Game -http://www.mrnussbaum.com/orderops/index.htmlYouTube - Order of Operations - Cartoon -http://www.youtube.com/watch?v=p14m2bDHTq8&feature=relatedProperties of OperationsSuite 101 - Teacher Tutorial -http://archive.suite101.com/article.cfm/math_fun/99844<strong>Math</strong> League - Properties - Teacher Tutorial -http://www.mathleague.com/help/wholenumbers/wholenumbers.htmPurplemath - Properties - Teacher Tutorial -http://www.purplemath.com/modules/numbprop.htmCoordinate Plane – Graphing Points in Quadrant INLVM - Counting All Pairs - Student Interactivehttp://nlvm.usu.edu/en/nav/frames_asid_307_g_4_t_1.html?from=category_g_4_t_1.htmlIXL - Location and Relative Coordinates on Maps - Assessment -http://www.ixl.com/math/grade-5/location-and-relative-coordinates-on-mapsIXL - Graph Points on a Coordinate Plane - Assessment -http://www.ixl.com/math/grade-5/graph-points-on-a-coordinate-planeIXL - Coordinate Graphs Review - Assessment -http://www.ixl.com/math/grade-5/coordinate-graphs-review-whole-numbersonlyUEN - Mountain Rescue Mission - Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=6168UEN - Fly on the Ceiling - Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=11237LearnAlberta - Ordered Pairs - Video Tutorial -http://www.learnalberta.ca/content/me5l/html/<strong>Math</strong>5.html?launch=trueEducation Place - Locate Points on a Grid - Student Tutorial -59[PLSD]

http://eduplace.com/cgibin/schtemplate.cgi?template=/math/hmm/models/tm_popup.thtml&grade=4&chapter=24&lesson=1&title=Locate+Points+on+a+Grid&tm=tmfe2401eOswego - Billy Bug - Game –http://www.oswego.org/ocsd-web/games/BillyBug/bugcoord.htmlEducation Place - Graphing on a Coordinate Grid - Student Tutorial -http://eduplace.com/cgibin/schtemplate.cgi?template=/math/hmm/models/tm_popup.thtml&grade=2&chapter=4&lesson=4&title=Graphing+on+a+Coordinate+Grid&tm=tmfc0404eUEN - “Fly on the Ceiling Lesson” -http://www.uen.org/Lessonplan/preview.cgi?LPid=11237Numerical PatternsTeacher’s Domain - Linking Number Patterns - Lesson -http://www.teachersdomain.org/resource/vtl07.math.algebra.pat.lpexponent/Teacher’s Domain - Finding the Common Beat - Lesson -http://www.teachersdomain.org/resource/vtl07.math.number.mul.commonbeat/UEN – “<strong>Math</strong> Stations for Pattern Review” Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=6164UEN - “Table Settings” Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=6159UEN - “Eye Spy a Rule” Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=15236WVPT4Learning - Problem Solving: Looking for a Pattern - Video -http://www.wvpt4learning.org/component/jomtube/video/426.html60[PLSD]

Unit 3: 5 th <strong>Grade</strong> Quarters2 and 3Approximately8 WeeksDomain: Measurement and Data and Number and OperationsStandards5.MD.3. Recognize volume as an attribute of solid figures and understandconcepts of volume measurement.a. A cube with side length 1 unit, called a “unit cube,” is said to have “onecubic unit” of volume, and can be used to measure volume.b. A solid figure which can be packed without gaps or overlaps using n unitcubes is said to have a volume of n cubic units.5.MD.4. Measure volumes by counting unit cubes, using cubic cm, cubic in,cubic ft, and improvised units.5.MD.5. Relate volume to the operations of multiplication and addition andsolve real world and mathematicalproblems involving volume.a. Find the volume of a right rectangular prism with whole-number sidelengths by packing it with unit cubes, and show that the volume is the same aswould be found by multiplying the edge lengths, equivalently by multiplyingthe height by the area of the base. Represent threefold whole-numberproducts as volumes, e.g., to represent the associative property ofmultiplication.b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms tofind volumes of right rectangular prisms with whole number edge lengths inthe context of solving real world and mathematical problems.c. Recognize volume as additive. Find volumes of solid figures composed oftwo non-overlapping right rectangular prisms by adding the volumes of thenon-overlapping parts, applying this technique to solve real world problems.5.NF.1. Add and subtract fractions with unlike denominators (includingmixed numbers) by replacing given fractions with equivalent fractions in sucha way as to produce an equivalent sum or difference of fractions with likedenominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general,a/b + c/d = (ad + bc)/bd.)5.NF.2. Solve word problems involving addition and subtraction of fractions61[PLSD]

PARCCPBA/MYAClarificationreferring to the same whole, including cases of unlike denominators, e.g., byusing visual fraction models or equations to represent the problem. Usebenchmark fractions and number sense of fractions to estimate mentally andassess the reasonableness of answers. For example, recognize an incorrectresult 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.5.MD.3i) Measures may include those in whole mm or cm.5.MD.4i) Tasks assess conceptual understanding of volume (see 5.MD.3) as appliedto a specific situation – not applying a volume formula.5.NF.1-1i) Tasks do not have a context.ii) Tasks ask for the answer or ask for an intermediate step that showsevidence of using equivalent fractions as a strategy.iii) Tasks do not include mixed numbers.iv) Tasks may involve fractions greater than 1 (including fractions and wholenumbers.)5.NF.2-1i) The situation types are those shown in Table 2,p. 9 of Progressions for Operations and Algebraic Thinking, sampled equallyacross rows and, within rows, sampled equally across columns.ii) Prompts do not provide visual fraction models; students may at theirdiscretion draw visual fraction models as a strategy.PARCCEOYClarification5.MD.3Recognize volume as an attribute of solid figures and understand concepts ofvolume measurement.a. A cube with side length 1 unit, called a “unit cube,” is said to have “onecubic unit” of volume, and can be used to measure volume.b. A solid figure which can be packed without gaps or overlaps using n&unitcubes is said to have a volume of n&cubic units.i) Tasks assess conceptual understanding of volume (see 5.MD.3) as appliedto a specific situation – not applying a volume formula.62[PLSD]

5.MD.4Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft,and improvised units.i) Tasks assess conceptual understanding of volume (see 5.MD.3) as appliedto a specific situation – not applying a volume formula.5.MD.5cRelate the operations of multiplication and addition and solve real world andmathematical problems involving volume.c. Recognize volume as additive. Find volumes of solid figures composed oftwo non-overlapping right rectangular prisms by adding the volumes of thenon- overlapping parts, applying this technique to solve real world problems.i) Tasks require students to solve a contextual problem by applying theindicated concepts and skills.5.MD.5dRelate volume to the operations of multiplication and addition and solve realworld and mathematical problems involving volume.b.Apply the formulas V= l × w × h and V=B x h&for&rectangular prisms tofind volumes of right rectangular prisms with whole-number edge lengths inthe context of solving real world and mathematical problems.i) Pool should contain tasks with and without contexts.ii) 50% of tasks involve use of V = l × w × h, 50% of tasks involve use ofV=B x h.iii) Tasks may require students to measure to find edge lengths to the nearestcm, mm or in.5.NF.1-163[PLSD]

5.NF.2-2i) The situation types are those shown in Table 2, p. 9 of Progression forOperations and Algebraic Thinking, sampled equally.ii) Prompts do not provide visual fraction models; students may at theirdiscretion draw visual fraction models as a strategy.5.NF.A.Int.1Solve word problems involving knowledge and skills articulated in 5.NF.1-1,5.NF.1-2, 5.NF.1-3, 5.NF.1-4, and 5.NF.1-5. “Prompts do not provide visualfraction models; students may at their discretion draw visual fraction modelsas a strategy”.SampleItems5.NF.1""Builds on the work in fourth grade where students add fractions with likedenominators. In fifth grade, the example provided in the standard 2/3 + 3⁄4has students find a common denominator by finding the product of bothdenominators. This process should come after students have used visualfraction models (area models, number lines, etc.) to build understandingbefore moving into the standard algorithm describes in the standard The useof these visual fraction models allows students to use reasonableness to find acommon denominator prior to using the algorithm. For example, when adding1/3 + 1/6, <strong>Grade</strong> 5 students should apply their understanding of equivalentfractions and their ability to rewrite fractions in an equivalent form to findcommon denominators.66[PLSD]

I drew a rectangle and shaded 1/3. I knew that if I cut every third in half thenI would have sixths. Based on my picture, 1/3 equals 2/6. Then I shaded inanother 1/6 with stripes. I ended up with an answer of 3/6, which is equal to1/2.On the contrary, based on the algorithm that is in the example of theStandard, when solving1/3 + 1/6, multiplying 3 and 6 gives a common denominator of 18. Studentswould make equivalent fractions 6/18 + 3/18 = 9/18 which is also equal toone-half. Please note that while multiplying the denominators will alwaysgive a common denominator, this may not result in the smallest denominator.Students should apply their understanding of equivalent fractions and theirability to rewrite fractions in an equivalent form to find commondenominators. They should know that multiplying the denominators willalways give a common denominator but may not result in the smallestdenominator.Examples:Fifth grade students will need to express both fractions in terms of a newdenominator with adding unlike denominators. For example, in calculating2/3 + 5/4 they reason that if each third in 2/3 is subdivided into fourths andeach fourth in 5/4 is subdivided into thirds, then each fraction will be a sumof unit fractions with denominator 3 x 4 = 4 x 3 + 12:It is not,necessary to find a least common denominator to calculate sums offractions, and in fact the effort of finding a least common denominator is adistraction from understanding adding fractions.67[PLSD]

(Progressions for&the&CCSSM,&Number&and&Operation&–&Fractions,&CCSSWriting Team, August 2011, page 10)Example:Present students with the problem 1/3 + 1/6. Encourage students to use theclock face as a model for solving the problem. Have students share theirapproaches with the class and demonstrate their thinking using the clockmodel.5.NF.2"This standard refers to number sense, which means students’ understandingof fractions as numbers that lie between whole numbers on a number line.Number sense in fractions also includes moving between decimals andfractions to find equivalents, also being able to use reasoning such as 7/8 isgreater than 3⁄4 because 7/8 is missing only 1/8 and 3⁄4 is missing 1⁄4 so 7/8is closer to a whole Also, students should use benchmark fractions to estimateand examine the reasonableness of their answers. Example here such as 5/8 isgreater than 6/10 because 5/8 is 1/8 larger than 1⁄2(4/8) and 6/10 is only 1/10larger than 1⁄2 (5/10)Example:Your teacher gave you 1/7 of the bag of candy. She also gave your friend 1/3of the bag of candy. If you and your friend combined your candy, whatfraction of the bag would you have? Estimate your answer and then calculate.How reasonable was your estimate?68[PLSD]

Example:Jerry was making two different types of cookies. One recipe needed 3/4 cupof sugar and the other needed 2/3 cup of sugar. How much sugar did he needto make both recipes?• Mental estimation:A student may say that Jerry needs more than 1 cup of sugar but lessthan 2 cups. An explanation may compare both fractions to 1⁄2 andstate that both are larger than 1⁄2 so the total must be more than 1. Inaddition, both fractions are slightly less than 1 so the sum cannot bemore than 2.• Area modelExample: Using a bar diagram• Sonia had 2 1/3 candy bars. She promised her brother that she wouldgive him 1⁄2 of a candy bar. How much will she have left after shegives her brother the amount she promised?• If Mary ran 3 miles every week for 4 weeks, she would reach her goalfor the month. The first day of the first week she ran 1 3⁄4 miles. Howmany miles does she still need to run the first week?o Using addition to find the answer:1 3⁄4 + n = 369[PLSD]

A student might add 11⁄4 to 13⁄4 to get to 3 miles. Then he or she would add1/6 more. Thus 11⁄4 miles + 1/6 of a mile is what Mary needs to run duringthat week.Example: Using an area model to subtract• This model shows 1 3⁄4 subtracted from 3 1/6 leaving 1 + 1⁄4 = 1/6which a student can then changeto 1 + 3/12 + 2/12 = 1 5/12. 3 1/6 can be expressed with a denominatorof 12. Once this is done a student can complete the problem, 2 14/12 –1 9/12 = 1 5/12.• This diagram models a way to show how 3 1/6 and 1 3⁄4 can beexpressed with a denominator of 12. Once this is accomplished, astudent can complete the problem, 2 14/12 – 1 9/12 = 1 5/12.Estimation skills include identifying when estimation is appropriate,determining the level of accuracy needed, selecting the appropriate method ofestimation, and verifying solutions or determining the reasonableness ofsituations using various estimation strategies. Estimation strategies forcalculations with fractions extend from students’ work with whole numberoperations and can be supported through the use of physical models.70[PLSD]

Example:Elli drank 3/5 quart of milk and Javier drank 1/10 of a quart less than Ellie.How much milk did they drink all together?Students make sense of fractional quantities when solving word problems,estimating answers mentally to see if they make sense.Example:Ludmilla and Lazarus each have a lemon. They need a cup of lemon juice tomake hummus for a party. Ludmilla squeezes 1/2 a cup from hers andLazarus squeezes 2/5 of a cup from his. How much lemon juice do they have?Is it enough?Students estimate that there is almost but not quite one cup of lemon juice,because 2/5 < 1/2. They calculate 1/2 + 2/5 = 9/10, and see this as 1/10 lessthan 1, which is probably a small enough shortfall that it will not ruin therecipe. They detect an incorrect result such as 2/5 + 2/5 = 3/7 by noticing that3/7 < 1/2.(Progressions for&the&CCSSM,&Number&and&Operation&–&Fractions,&CCSSWriting Team, August 2011, page 11)5. MD.3, 5.MD.4, and 5. MD.5These standards represent the first time that students begin exploring theconcept of volume. In third grade, students begin working with area andcovering spaces. The concept of volume should be extended from area withthe idea that students are covering an area (the bottom of cube) with a layer ofunit cubes and then adding layers of unit cubes on top of bottom layer (seepicture below). Students should have ample experiences with concretemanipulatives before moving to pictorial representations. Students’ priorexperiences with volume were restricted to liquid volume. As studentsdevelop their understanding volume they understand that a 1-unit by 1-unit by71[PLSD]

1-unit cube is the standard unit for measuring volume. This cube has a lengthof 1 unit, a width of 1 unit and a height of 1 unit and is called a cubic unit.This cubic unit is written with an exponent of 3 (e.g., in3, m3). Studentsconnect this notation to their understanding of powers of 10 in our placevalue system. Models of cubic inches, centimeters, cubic feet, etc are helpfulin developing an image of a cubic unit. Students’ estimate how many cubicyards would be needed to fill the classroom or how many cubic centimeterswould be needed to fill a pencil box.The major emphasis for measurement in <strong>Grade</strong> 5 is volume. Volume not onlyintroduces a third dimension and thus a significant challenge to students’spatial structuring, but also complexity in the nature of the materialsmeasured. That is, solid units are “packed,” such as cubes in a threedimensionalarray, whereas a liquid “fills” three-dimensional space, takingthe shape of the container. The unit structure for liquid measurement may bepsychologically one dimensional for some students.“Packing” volume is more difficult than iterating a unit to measure length andmeasuring area by tiling. Students learn about a unit of volume, such as acube with a side length of 1 unit, called a unit cube.5.MD.3 They pack cubes(without gaps) into right rectangular prisms and count the cubes to determinethe volume or build right rectangular prisms from cubes and see the layers asthey build.5.MD.4 They can use the results to compare the volume of rightrectangular prisms that have different dimensions. Such experiences enablestudents to extend their spatial structuring from two to three dimensions. Thatis, they learn to both mentally decompose and recompose a right rectangularprism built from cubes into layers, each of which is composed of rows andcolumns. That is, given the prism, they have to be able to decompose it,understanding that it can be partitioned into layers, and each layer partitionedinto rows, and each row into cubes. They also have to be able to composesuch as structure, multiplicatively, back into higher units. That is, theyeventually learn to conceptualize a layer as a unit that itself is composed of72[PLSD]

units of units—rows, each row composed of individual cubes—and theyiterate that structure. Thus, they might predict the number of cubes that willbe needed to fill a box given the net of the box.Another complexity of volume is the connection between “packing” and“filling.” Often, for example, students will respond that a box can be filledwith 24 centimeter cubes, or build a structure of 24 cubes, and still think ofthe 24 as individual, often discrete, not necessarily units of volume. Theymay, for example, not respond confidently and correctly when asked to fill agraduated cylinder marked in cubic centimeters with the amount of liquid thatwould fill the box. That is, they have not yet connected their ideas aboutfilling volume with those concerning packing volume. Students learn to movebetween these conceptions, e.g., using the same container, both filling (from agraduated cylinder marked in ml or cc) and packing (with cubes that are each1 cm3). Comparing and discussing the volume-units and what they representcan help students learn a general, complete, and interconnectedconceptualization of volume as filling three-dimensional space.Students then learn to determine the volumes of several right rectangularprisms, using cubic centimeters, cubic inches, and cubic feet. With guidance,they learn to increasingly apply multiplicative reasoning to determinevolumes, looking for and making use of structure. That is, they understandthat multiplying the length times the width of a right rectangular prism can beviewed as determining how many cubes would be in each layer if the prismwere packed with or built up from unit cubes.5.MD.5a They also learn thatthe height of the prism tells how many layers would fit in the prism. That is,they understand that volume is a derived attribute that, once a length unit isspecified, can be computed as the product of three length measurements or asthe product of one area and one length measurement.Then, students can learn the formulas V =l x w x h and V = B x h for rightrectangular prisms as efficient methods for computing volume, maintainingthe connection between these methods and their previous work withcomputing the number of unit cubes that pack a right rectangularprism.5.MD.5b They use these competencies to find the volumes of rightrectangular prisms with edges whose lengths are whole numbers and solvereal-world and mathematical problems involving such prisms.Students also recognize that volume is additive and they find the total volumeof solid figures composed of two right rectangular prisms.5.MD.5c Forexample, students might design a science station for the ocean floor that is73[PLSD]

composed of several rooms that are right rectangular prisms and that meet aset criterion specifying the total volume of the station. They draw their stationand justify how their design meets the criterion.5. MD.5a & b These standards involve finding the volume of rightrectangular prisms (see picture above). Students should have experiences todescribe and reason about why the formula is true. Specifically, that they arecovering the bottom of a right rectangular prism (length x width) withmultiple layers (height). Therefore, the formula (length x width x height) is anextension of the formula for the area of a rectangle.5.MD.5c This standard calls for students to extend their work with the area ofcomposite figures into the context of volume. Students should be givenconcrete experiences of breaking apart (decomposing) 3-dimensional figuresinto right rectangular prisms in order to find the volume of the entire 3-dimensional figure.74[PLSD]

Students need multiple opportunities to measure volume by filling rectangularprisms with cubes and looking at the relationship between the total volumeand the area of the base. They derive the volume formula (volume equals thearea of the base times the height) and explore how this idea would apply toother prisms. Students use the associative property of multiplication anddecomposition of numbers using factors to investigate rectangular prismswith a given number of cubic units.75[PLSD]

Example:When given 24 cubes, students make as many rectangular prisms as possiblewith a volume of 24 cubic units. Students build the prisms and recordpossible dimensions.Example:Students determine the volume of concrete needed to build the steps in thediagram below.I CanStatements5.MD.3• I can understand volume5.MD.4• I can measure volume by counting unit cubes5.MD.5• I can solve real world problems involving volume76[PLSD]

5.NF.1• I can add & subtract fraction with like denominators.• I can find multiples of a given number.• I can find common multiples of a set of numbers.• I can find least common multiples of a set of numbers. (LCD)• I can add & subtract fractions with unlike denominators. (Simplestform)5.NF.2• I can solve word problems using addition & subtraction of fractionswith unlike denominators.• I can use fraction models to solve addition & subtraction of fractions.• I can use benchmark fractions to estimate fractions mentally.• I can check that my answer is reasonable.Vocabulary• addend• Associative Property of Addition• benchmark fractions• common denominators• common multiples• Commutative Property of Addition• denominator• difference• equivalent fractions• estimate• fraction• fraction greater than 1• fraction less than 1• like denominators• lowest terms• minuend• mixed number• numerator• reasonableness• simplest form• simplify• subtrahend• sum77[PLSD]

InstructionalStrategies• unlike denominator• Associative Property of Multiplication• attribute• base of a solid figure• congruent• cubic unit• decagon• decagonal prism• formula• hexagonal prism• lateral face• octagon• octagonal prism• pentagonal prism• pentagonal pyramid• polygon• polyhedron• prism• pyramid• right rectangular prism• solid figure• three-dimensional figures• unit cube• volume5.MD.3, 5.MD.4 AND 5.MD5Instructional StrategiesVolume refers to the amount of space that an object takes up and is measuredin cubic units such as cubic inches or cubic centimeters.Students need to experience finding the volume of rectangular prisms bycounting unit cubes, in metric and standard units of measure, before theformula is presented. Provide multiple opportunities for students to developthe formula for the volume of a rectangular prism with activities similar to theone described below.Give students one block (a I- or 2- cubic centimeter or cubic-inch cube), aruler with the appropriate measure based on the type of cube, and a smallrectangular box. Ask students to determine the number of cubes needed to fillthe box. Have students share their strategies with the class using words,78[PLSD]

drawings or numbers. Allow them to confirm the volume of the box by fillingthe box with cubes of the same size.By stacking geometric solids with cubic units in layers, students can beginunderstanding the concept of how addition plays a part in finding volume.This will lead to an understanding of the formula for the volume of a rightrectangular prism, b x h, where b is the area of the base. A right rectangularprism has three pairs of parallel faces that are all rectangles.Have students build a prism in layers. Then, have students determine thenumber of cubes in the bottom layer and share their strategies. Studentsshould use multiplication based on their knowledge of arrays and its use inmultiplying two whole numbers.Ask what strategies can be used to determine the volume of the prism basedon the number of cubes in the bottom layer. Expect responses such as “addingthe same number of cubes in each layer as were on the bottom layer” ormultiply the number of cubes in one layer times the number of layers.Instructional Resources/ToolsCubesRulers (marked in standard or metric units) Grid paperhttp://illuminations.nctm.org/ActivityDetail.aspx?ID=6 - Determining theVolume of a Box by Filling It with Cubes, Rows of Cubes, or Layers ofCubes5.NF.1 AND 5.NF.2Instructional StrategiesTo add or subtract fractions with unlike denominators, students use theirunderstanding of equivalent fractions to create fractions with the samedenominators. Start with problems that require the changing of one of thefractions and progress to changing both fractions. Allow students to add andsubtract fractions using different strategies such as number lines, area models,fraction bars or strips. Have students share their strategies and discuss79[PLSD]

epresentations need to be from the same whole models with the same shapeand size.ConnectionsThis cluster is connected to the <strong>Grade</strong> 5 Critical Area of Focus #3,Developing understanding of volume.CRITICAL AREA OF FOCUS #3Developing understanding of volume. Students recognize volume as anattribute of three-dimensional space. They understand that volume can bemeasured by finding the total number of same-size units of volume requiredto fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They selectappropriate units, strategies, and tools for solving problems that involveestimating and measuring volume. They decompose three-dimensional shapesand find volumes of right rectangular prisms by viewing them as decomposedinto layers of arrays of cubes. They measure necessary attributes of shapes inorder to determine volumes to solve real world and mathematical problems.ResourcesUse place value understanding and properties of operations to perform multidigitarithmetic (<strong>Grade</strong> 4 NBT 5).Equivalent FractionsLearn Alberta - Equivalent Fractions- Video Tutorial -http://www.learnalberta.ca/content/me5l/html/<strong>Math</strong>5.html?launch=trueEducation Place - Equivalent Fractions and Simplest Form - Student Tutorialfile://localhost/- http/::eduplace.com:cgibin:schtemplate.cgi%3Ftemplate=:math:hmm:models:tm_popup.thtml&grade=5&chapter=9&lesson=6&title=Equivalent+Fractions+and+Simplest+Form&tm=tm ff0906e81[PLSD]

IXL - Volume of Cubes and Rectangular Prisms - Assessment -http://www.ixl.com/math/grade-5/volumeLearn Alberta - Volume - Video Tutorial -http://www.learnalberta.ca/content/me5l/html/<strong>Math</strong>5.html?launch=trueScholastic Study Jams - Volume - Student Tutorial -http://studyjams.scholastic.com/studyjams/jams/math/measurement/volume.htmIlluminations - Fill ‘er Up Lesson -http://illuminations.nctm.org/LessonDetail.aspx?id=L831Illuminations - “Fishing for the Best Prism” Lesson -http://illuminations.nctm.org/LessonDetail.aspx?id=L793Illuminations - “Popcorn, Anyone?” Lesson -http://illuminations.nctm.org/LessonDetail.aspx?id=L797LearnAlberta - Volume and Displacement - Lesson -http://www.learnalberta.ca/content/mesg/html/math6web/index.html?page=lessons&lesson=m6lessonshell15.swfThree-Dimensional Box - Working with Volume - Applet -http://mste.illinois.edu/users/carvell/3dbox/default.html<strong>Math</strong>Open Reference - Interactive Model -http://www.mathopenref.com/cubevolume.htmlUEN - “Box It Up” Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=2154583[PLSD]

Unit 4: 5 th <strong>Grade</strong> Quarters3 and 4StandardsDomain: Number and Operations84[PLSD]Approximately8 Weeks5.NF.3. Interpret a fraction as division of the numerator by the denominator(a/b = a ÷ b). Solve word problems involving division of whole numbersleading to answers in the form of fractions or mixed numbers, e.g., by usingvisual fraction models or equations to represent the problem.For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4people 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 eachperson get? Between what two whole numbers does your answer lie?5.NF.4. Apply and extend previous understandings of multiplication tomultiply a fraction or whole number by a fraction.a. Interpret the product (a/b) × q as a parts of a partition of q into b equalparts; equivalently, as the result of a sequence of operations a × q ÷ b.For example, use a visual fraction model to show (2/3) × 4 = 8/3, and createa story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (Ingeneral, (a/b) × (c/d) = ac/bd.)b. Find the area of a rectangle with fractional side lengths by tiling it with unitsquares of the appropriate unit fraction side lengths, and show that the area isthe same as would be found by multiplying the side lengths. Multiplyfractional side lengths to find areas of rectangles, and represent fractionproducts as rectangular areas.5.NF5. Interpret multiplication as scaling (resizing), by:a. Comparing the size of a product to the size of one factor on the basis of thesize of the otherfactor, without performing the indicated multiplication.b. Explaining why multiplying a given number by a fraction greater than 1results in a product greater than the given number (recognizing multiplicationby whole numbers greater than 1 as a familiar case); explaining whymultiplying a given number by a fraction less than 1 results in a productsmaller than the given number; and relating the principle of fraction

equivalence a/b =(n×a)/(n×b) to the effect of multiplying a/b by 1.5.NF.6. Solve real world problems involving multiplication of fractions andmixed numbers, e.g., by using visual fraction models or equations torepresent the problem.5.NF.7. Apply and extend previous understandings of division to divide unitfractions by whole numbers and whole numbers by unit fractions.a. Interpret division of a unit fraction by a non-zero whole number, andcompute such quotients. For example, create a story context for (1/3) ÷ 4, anduse a visual fraction model to show the quotient. Use the relationship betweenmultiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4= 1/3.b. Interpret division of a whole number by a unit fraction, and compute suchquotients. For example, create a story context for 4 ÷ (1/5), and use a visualfraction model to show the quotient. Use the relationship betweenmultiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) =4.c. Solve real world problems involving division of unit fractions by non-zerowhole numbers and division of whole numbers by unit fractions, e.g., byusing visual fraction models and equations to represent the problem. Forexample, how much chocolate will each person get if 3 people share 1/2 lb ofchocolate equally? How many 1/3-cup servings are in 2 cups of raisins?PARCCPBA/MYAClarification5.MD.2. Make a line plot to display a data set of measurements in fractions ofa unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solveproblems involving information presented in line plots. For example, givendifferent measurements of liquid in identical beakers, find the amount ofliquid each beaker would contain if the total amount in all the beakers wereredistributed equally.5.NF.3i) Tasks do not have a context.5.NF.3-2i) Prompts do not provide visual fraction models; students may at theirdiscretion draw visual fraction models as a strategy.ii) Note that one of the italicized examples in standard 5.NF.3 is a twopromptproblem.85[PLSD]

how many pounds of rice should each person get? Between what two wholenumbers does your answer lie?i) Prompts do not provide visual fraction models; students may at theirdiscretion draw visual fraction models as a strategy.ii) Note that one of the italicized examples in standard 5.NF.3 is a twopromptproblem.5.NF.4a-1i) Tasks require finding a fractional part of a whole number quantity.ii) The result is equal to a whole number in 20% of tasks; these are practiceforwardfor MP.7.iii) Tasks have “thin context” or no context.5.NF.4a-287[PLSD]

i) Tasks require finding a product of two fractions (neither of the factorsequal to a whole number).ii) The result is equal to a whole number in 20% of tasks; these are practiceforwardfor MP.7.iii) Tasks have “thin context” or no context.5.NF.4b-1Apply and extend previous understandings of multiplication to multiply afraction or whole number by a fraction.a. Multiply fractional side lengths to find areas of rectangles, and representfraction products as rectangular areas.i) 50% of the tasks present students with the rectangle dimensions and askstudents to find the area; 50% of the tasks give the fractions and the productand ask students to show a rectangle to model the problem.5.NF.5aInterpret multiplication as scaling (resizing), bya. Comparing the size of a product to the size of one factor on the basis of thesize of the other factor, without performing the indicated multiplication.i) Insofar as possible, tasks are designed to be completed without performingthe indicated multiplication.ii) Products involve at least one factor that is a fraction or mixed number.5.NF.6-1Solve real world problems involving multiplication of fractions, e.g., by usingvisual fraction models or equations to represent the problem.i) Tasks do not involve mixed numbers.ii) Situations include area and comparison/times as much, with productunknown. (See Table 2, p. 89 of CCSS and Table 3, p. 23 of Progression forOperations and Algebraic Thinkingiii) Prompts do not provide visual fraction models; students may at theirdiscretion draw visual fraction models as a strategy.5.NF.6-2Solve real world problems involving multiplication of fractions and mixednumbers, e.g., by using visual fraction models or equations to represent theproblem.88[PLSD]

i) Tasks present one or both factors in the form of a mixed number.ii) Situations include area and comparison/times as much, with productunknown.iii) Prompts do not provide visual fraction models; students may at theirdiscretion draw visual fraction models as a strategy.5.NF.7a5.NF.7b5.NF.7bc. Solve real world problems involving division of unit fractions by non-zerowhole numbers and division of whole numbers by unit fractions, e.g., byusing visual fraction models and equations to represent the problem. Forexample, how much chocolate will each person get if 3 people share 1/2 lb ofchocolate equally? How many 1/3-cup servings are in 2 cups of raisins?i) Tasks involve equal group (partition) situations with part size unknown andnumber of parts unknown. (See Table 2, p. 89, CCSS)89[PLSD]

ii) Prompts do not provide visual fraction models; students may at theirdiscretion draw visual fraction models as a strategy.5.MD.2-1Make a line plot to display a data set of measurements in fractions of a unit(1/2, 1/4, 1/8).5.MD.2-2Use operations on fractions for this grade (knowledge and skills articulated in5.NF) to solve problems involving information in line plots. For example,given different measurements of liquid in identical beakers, find the amountof liquid each beaker would contain if the total amount in all the beakerswere redistributed equally.SampleItems5.NF.3Fifth grade student should connect fractions with division, understanding that5 ÷ 3 = 5/3 Students should explain this by working with their understandingof division as equal sharing.90[PLSD]

Students should also create story contexts to represent problems involvingdivision of whole numbers.Example:If 9 people want to share a 50-pound sack of rice equally by weight, howmany pounds of rice should each person get? This can be solved in two ways.First, they might partition each pound among the 9 people, so that eachperson gets50 x 1/9 = 50/9 pounds.Second, they might use the equation 9 x 5= 45 to see that each person can begiven 5 pounds, with 5 pounds remaining. Partitioning the remainder gives 55/9 pounds for each person.This standard calls for students to extend their work of partitioning a numberline from third and fourth grade. Students need ample experiences to explorethe concept that a fraction is a way to represent the division of two quantities.Students are expected to demonstrate their understanding using concretematerials, drawing models, and explaining their thinking when working withfractions in multiple contexts. They read 3/5 as “three fifths” and after manyexperiences with sharing problems, learn that 3/5 can also be interpreted as “3divided by 5.”Examples:Ten team members are sharing 3 boxes of cookies. How much of a box willeach student get?When working this problem a student should recognize that the 3 boxes arebeing divided into 10 groups, so s/he is seeing the solution to the followingequation, 10 x n = 3 (10 groups of some amount is 3 boxes) which can also bewritten as n = 3 ÷ 10. Using models or diagram, they divide each box into 10groups, resulting in each team member getting 3/10 of a box.Two afterschool clubs are having pizza parties. For the <strong>Math</strong> Club, theteacher will order 3 pizzas for every 5 students. For the student council, theteacher will order 5 pizzas for every 8 students. Since you are in both groups,you need to decide which party to attend. How much pizza would you get ateach party? If you want to have the most pizza, which party should you91[PLSD]

attend?The six fifth grade classrooms have a total of 27 boxes of pencils. How manyboxes will each classroom receive? Students may recognize this as a wholenumber division problem but should also express this equal sharing problemas 27/6They explain that each classroom gets 27/6 boxes of pencils and can furtherdetermine that each classroom get 4 3/6 or 4 ½ boxes of pencils.Example:Your teacher gives 7 packs of paper to your group of 4 students. If you sharethe paper equally, how much paper does each student get?Each student receives 1 whole pack of paper and 1⁄4 of the each of the 3packs of paper. So each student gets 1 3⁄4 packs of paper.5.NF.4Students need to develop a fundamental understanding that the multiplicationof a fraction by a whole number could be represented as repeated addition ofa unit fraction (e.g., 2 x (1/4) = 1/4 + 1⁄4This standard extends student’s work of multiplication from earlier grades. Infourth grade, students worked with recognizing that a fraction such as 3/5actually could be represented as 3 pieces that are each one-fifth (3 x (1/5)).This standard references both the multiplication of a fraction by a wholenumber and the multiplication of two fractions. Visual fraction models (areamodels, tape diagrams, number lines) should be used and created by studentsduring their work with this standard.92[PLSD]

As they multiply fractions such as 3/5 x 6, they can think of the operation inmore than one way.• 3 x (6 ÷ 5) or (3 x 6/5)• (3 x 6) ÷ 5 or 18 ÷ 5 (18/5)Students create a story problem for 3/5 x 6 such as,• Isabel had 6 feet of wrapping paper. She used 3/5 of the paper to wrapsome presents. How much does she have left?• Every day Tim ran 3/5 of mile. How far did he run after 6 days?(Interpreting this as 6 x 3/5)Example:Three-fourths of the class is boys. Two-thirds of the boys are wearing tennisshoes. What fraction of the class are boys with tennis shoes?This question is asking what 2/3 of 3⁄4 is, or what is 2/3 x 3⁄4. What is 2/3 x3⁄4, in this case you have 2/3 groups of size 3⁄4 (a way to think about it interms of the language for whole numbers is 4 x 5 you have 4 groups of size 5.The array model is very transferable from whole number work and then tobinomials.93[PLSD]

5.NF.5This standard calls for students to examine the magnitude of products interms of the relationship between two types of problems. This extends thework with 5.OA.1.This standard asks students to examine how numbers change when wemultiply by fractions. Students should have ample opportunities to examineboth cases in the standard: a) when multiplying by a fraction greater than 1,the number increases and b) when multiplying by a fraction less the one, thenumber decreases. This standard should be explored and discussed whilestudents are working with 5.NF.4, and should not be taught in isolation.95[PLSD]

Example:Mrs. Bennett is planting two flower beds. The first flower bed is 5 meterslong and 6/5 meters wide. The second flower bed is 5 meters long and 5/6meters wide. How do the areas of these two flower beds compare? Is thevalue of the area larger or smaller than 5 square meters? Draw pictures toprove your answer.5.NF.6This standard builds on all of the work done in this cluster. Students shouldbe given ample opportunities to use various strategies to solve word problemsinvolving the multiplication of a fraction by a mixed number. This standardcould include fraction by a fraction, fraction by a mixed number or mixednumber by a mixed number.Example:There are 2 1⁄2 bus loads of students standing in the parking lot. The studentsare getting ready to go on a field trip. 2/5 of the students on each bus are girls.How many busses would it take to carry only the girls?96[PLSD]

5.NF.7 is the first time that students are dividing with fractions. In fourthgrade students divided whole numbers, and multiplied a whole number by afraction. The concept unit fraction is a fraction that has a one in thedenominator. For example,the fraction 3/5 is 3 copies of the unit fraction1/5.1/5 +1/5+1/5 =3/5=1/5 x3or3x1/5Example:Knowing the number of groups/shares and finding how many/much in eachgroup/share. Four students sitting at a table were given 1/3 of a pan ofbrownies to share. How much of a pan will each student get if they share thepan of brownies equally?97[PLSD]

The diagram shows the 1/3 pan divided into 4 equal shares with each shareequaling 1/12 of the pan.5.NF.7aThis standard asks students to work with story contexts where a unit fractionis divided by a non-zero whole number. Students should use various fractionmodels and reasoning about fractions.Example:You have 1/8 of a bag of pens and you need to share them among 3 people.How much of the bag does each person get?98[PLSD]

5.NF.7b This standard calls for students to create story contexts and visualfraction models for division situations where a whole number is being dividedby a unit fraction.Example:Create a story context for 5 ÷ 1/6. Find your answer and then draw a pictureto prove your answer and use multiplication to reason about whether youranswer makes sense. How many 1/6 are there in 5?5.NF.7c extends students’ work from other standards in 5.NF.7. Studentshould continue to use visual fraction models and reasoning to solve thesereal-world problems.Example:How many 1/3-cup servings are in 2 cups of raisins?99[PLSD]

5.MD.2This standard provides a context for students to work with fractions bymeasuring objects to one-eighth of a unit. This includes length, mass, andliquid volume. Students are making a line plot of this data and then addingand subtracting fractions based on data in the line plot.Example:Students measured objects in their desk to the nearest 1⁄2, 1⁄4, or 1/8 of aninch then displayed data collected on a line plot. How many object measured1⁄4? 1⁄2? If you put all the objects together end to end what would be the totallength of all the objects?Example:Ten beakers, measured in liters, are filled with a liquid.The line plot above shows the amount of liquid in liters in 10 beakers. If the100[PLSD]

liquid is redistributed equally, how much liquid would each beaker have?(This amount is the mean.)Students apply their understanding of operations with fractions. They useeither addition and/or multiplication to determine the total number of liters inthe beakers. Then the sum of the liters is shared evenly among the tenbeakers.I CanStatements5.NF.3• I understand that a fraction represents a division problem.• I can represent a division problem as a fraction.• I can solve division word problems whose quotients are fractions ormixed numbers.5.NF.4• I can multiply a fraction by a whole number• I can multiply a fractions by a fraction.• I can show my answer with a fraction model.• I can create a story for this equation.• I can explain the product of a whole number and a fraction.• I can show the product of a whole number and a fraction on a grid.• I can use tiling to prove the product of a fraction equation.• I can show fraction products as rectangular areas.5.NF.5• I can determine a reasonable product based on the size of the factorsbefore multiplying.• I can explain why a number multiplied by a fraction less than 1 resultsin a product less than the given number.• I can explain why a number multiplied by a fraction greater than 1results in a product greater than the given number.5.NF.6• I can solve real-world problems involving multiplication of fractions &mixed numbers using visual fraction models.5.NF.7• I can divide a fraction by a whole number.• I can show my answer with a fraction model.• I can create a story for this equation.• I can use the relationship between multiplication & division to explainthe equation.101[PLSD]

• I can divide a whole number by a fraction.• I can show my answer with a fraction model.• I can create a story for this equation.• I can use the relationship between multiplication & division to explainthe equation.• I can solve real-world problems involving division of fractions by awhole number using visual fraction models.• I can solve real-world problems involving division of a whole numberby a fraction using visual fraction models.5.MD.2• I can use fraction operations to solve problems involving informationpresented on a line plot.Vocabulary• compatible numbers• denominator• dividend• divisor• equation• estimate• fraction bar• long division• mixed number• numerator• partial quotients• quotient• remainder• area• array• denominator• equation• equivalent fractions• factor• fraction greater than 1• fraction less than 1• mixed number• Multiplicative Identity Property of 1• numerator• product102[PLSD]

InstructionalStrategies• rectangle• scaling• simplest form• simplify• square unit• tiling• whole numbers• axis (axes)• coordinate plane• coordinate system• coordinates• corresponding terms• data• fraction• intersect• interval• line graph• line plotInstructional StrategiesConnect the meaning of multiplication and division of fractions with wholenumbermultiplication and division. Consider area models of multiplicationand both sharing and measuring models for division.Encourage students to use models or drawings to multiply or divide withfractions. Begin with students modeling multiplication and division withwhole numbers. Have them explain how they used the model or drawing toarrive at the solution.Models to consider when multiplying or dividing fractions include, but arenot limited to: area models using rectangles or squares, fraction strips/barsand sets of counters.103[PLSD]

Present problem situations and have students use models and equations tosolve the problem. It is important for students to develop understanding ofmultiplication and division of fractions through contextual situations.Instructional Resources/ToolsThe National Library of Virtual Manipulatives: contains Java applets andactivities for K-12 mathematics.http://nlvm.usu.edu/en/nav/vlibrary.htmlFractions - Rectangle Multiplication – students can visualize and practicemultiplying fractions using an area representation.http://nlvm.usu.edu/en/nav/frames_asid_194_g_2_t_1.htmlNumber Line Bars– Fractions: students can divide fractions using numberline bars.http://nlvm.usu.edu/en/nav/frames_asid_180_g_2_t_1.htmlORC # 5812 Divide and Conquer - Students can better understand thealgorithm for dividing fractions if they analyze division through a sequenceof problems starting with division of whole numbers, followed by division ofa whole number by a unit fraction, division of a whole number by a non-unitfraction, and finally division of a fraction by a fraction (addressed in <strong>Grade</strong>6).http://www.utdanacenter.org/mathtoolkit/instruction/lessons/7_divide.phpCommon MisconceptionsStudents may believe that multiplication always results in a larger number.Using models when multiplying with fractions will enable students to see that104[PLSD]

the results will be smaller.ConnectionsAdditionally, students may believe that division always results in a smallernumber. Using models when dividing with fractions will enable students tosee that the results will be larger.This cluster is connected to the <strong>Grade</strong> 5 Critical Area of Focus #1,Developing fluency with addition and subtraction of fractions and developingunderstanding of the multiplication of fractions and of division of fractions inlimited cases (unit fractions divided by whole numbers and whole numbersdivided by unit fractions).CRITICAL AREA OF FOCUS #1Developing fluency with addition and subtraction of fractions and developingunderstanding of the multiplication of fractions and of division of fractions inlimited cases (unit fractions divided by whole numbers and whole numbersdivided by unit fractions)Students apply their understanding of fractions and fraction models torepresent the addition and subtraction of fractions with unlike denominatorsas equivalent calculations with like denominators. They develop fluency incalculating sums and differences of fractions, and make reasonable estimatesof them. Students also use the meaning of fractions, of multiplication anddivision, and the relationship between multiplication and division tounderstand and explain why the procedures for multiplying and dividingfractions make sense. (Note: this is limited to the case of dividing unitfractions by whole numbers and whole numbers by unit fractions.)ResourcesMultiplying FractionsNLVM - Rectangle Multiplication of Fractions - Interactive Applet -http://nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=category_g_3_t_1.html<strong>Math</strong> Is Fun - Multiplying Fractions - Student Tutorial -http://www.mathsisfun.com/fractions_multiplication.html<strong>Math</strong> Playground - Multiplying Fractions - Interactive Applet -http://www.mathplayground.com/fractions_mult.html<strong>Math</strong> Is Fun - Multiplying Mixed Numbers - Student Tutorial -105[PLSD]

http://www.mathsisfun.com/mixed-fractions-multiply.htmlYouTube - Multiplying Mixed Numbers - Teacher Tutorial -http://www.youtube.com/watch?v=cDg5_Ft9SZs<strong>Math</strong> Play - Multiplying Fractions Millionaire Game - Game -http://www.math-play.com/Multiplying-Fractions-Millionaire/Multiplying-Fractions- Millionaire.htmlDivision of Fractions with a Whole NumberIXL - Divide Fractions by Whole Numbers - Assessment -http://www.ixl.com/math/grade-5/divide-fractions-by-whole-numbersIXL- Divide Whole Numbers by Fractions - Assessment -http://www.ixl.com/math/grade-5/divide-whole-numbers-by-fractionsUEN - “Fruity O Fractions” Lesson -http://www.uen.org/Lessonplan/preview.cgi?LPid=6156General Line Plot InformationIXL - Create Line Plots - Assessment –http://www.ixl.com/math/grade-6/create-line-plotsLearnAlberta - Displaying Data - Video Tutorial -http://www.learnalberta.ca/content/me5l/html/math5.html?goLesson=21IXL - Interpret Line Plots - Assessment –http://www.ixl.com/math/grade-5/interpret-line-plots106[PLSD]

Unit 5: 5 th <strong>Grade</strong> Quarter4Domain: GeometryStandardsApproximately2 Weeks5.G.3. Understand that attributes belonging to a category of two dimensionalfigures also belong to all subcategories of that category.For example, all rectangles have four right angles and squares arerectangles, so all squares have four right angles.PARCCEOYClarification5.G.4. Classify two-dimensional figures in a hierarchy based on properties.5.G.3Understand that attributes belonging to a category of two- dimensionalfigures also belong to all subcategories of that category. For example, allrectangles have four right angles and squares are rectangles, so all squareshave four right angles.i) A trapezoid is defined as “A quadrilateral with at least one pair of parallelsides.”5.G.4Classify two-dimensional figures in a hierarchy based on properties.A trapezoid is defined as “A quadrilateral with at least one pair of parallelsides.”SampleItems5.G.3This standard calls for students to reason about the attributes (properties) ofshapes. Student should have experiences discussing the property of shapesand reasoning.Example:Examine whether all quadrilaterals have right angles. Give examples andnon-examples.Example:If the opposite sides on a parallelogram are parallel and congruent, thenrectangles are parallelogramsA sample of questions that might be posed to students include:A parallelogram has 4 sides with both sets of opposite sides parallel. What107[PLSD]

types of quadrilaterals are parallelograms? Regular polygons have all of theirsides and angles congruent. Name or draw some regular polygons.All rectangles have 4 right angles. Squares have 4 right angles so they arealso rectangles. True or False?A trapezoid has 2 sides parallel so it must be a parallelogram. True or False?The notion of congruence (“same size and same shape”) may be part ofclassroom conversation but the concepts of congruence and similarity do not,appear until middle school.TEACHER,NOTE:,In the U.S., the term “trapezoid” may have two differentmeanings. Research identifies these as inclusive and exclusive definitions.The inclusive definition states: A trapezoid is a quadrilateral with at&least&onepair of parallel sides. The exclusive definition states: A,trapezoid,is,a,quadrilateral,with,exactly(one,pair,of,parallel,sides.,With thisdefinition, a parallelogram is not a trapezoid. North Carolina has adopted theexclusive definition. (Progressions for&the&CCSSM:&Geometry,&The CommonCore Standards Writing Team, June 2012.)http://illuminations.nctm.org/ActivityDetail.aspx?ID=705.G.4This standard builds on what was done in 4th grade.Figures from previous grades: polygon,,rhombus/rhombi,,rectangle,,square,,triangle,,quadrilateral,,pentagon,,hexagon,,cube,,trapezoid,,half/quarter,circle,,circle,,kite,,A kite,is a quadrilateral whose four sides can be grouped into two pairs ofequal-length sides that are beside (adjacent to) each other.108[PLSD]

Example:Create a Hierarchy Diagram using the following terms:Student should be able to reason about the attributes of shapes by examining:What are ways to classify triangles? Why can’t trapezoids and kites beclassified as parallelograms? Which quadrilaterals have opposite anglescongruent and why is this true of certain quadrilaterals?, and How many linesof symmetry does a regular polygon have?TEACHER NOTE: In the U.S., the term “trapezoid” may have two differentmeanings. Research identifies these as inclusive and exclusive definitions.The inclusive definition states: A trapezoid is a quadrilateral with at least onepair of parallel sides. The exclusive definition states:A trapezoid is a quadrilateral with exactly one pair of parallel sides. Withthis definition, a parallelogram is not a trapezoid. North Carolina has adoptedthe exclusive definition. (Progressions for the CCSSM: Geometry, TheCommon Core Standards Writing Team, June 2012.)109[PLSD]

I CanStatements5.G.3• I can classify shapes into categories.5.G.4• I can classify shapes based on properties.Vocabulary• acute triangle• attribute• congruent• decagon• equilateral triangle• formula• isosceles triangle• hierarchy• heptagon• hexagon• nonagon• obtuse triangle• octagon• parallel lines• parallelogram• pentagon• perpendicular lines• polygon• quadrilateral• rectangle• regular polygon• rhombus• right rectangular prism• right triangle• scalene triangle• trapezoid• two-dimensional figuresInstructionalStrategiesInstructional StrategiesThis cluster builds from <strong>Grade</strong> 3 when students described, analyzed andcompared properties of two-dimensional shapes. They compared andclassified shapes by their sides and angles, and connected these withdefinitions of shapes. In <strong>Grade</strong> 4 students built, drew and analyzed two-110[PLSD]

dimensional shapes to deepen their understanding of the properties of twodimensionalshapes. They looked at the presence or absence of parallel andperpendicular lines or the presence or absence of angles of a specified size toclassify two-dimensional shapes. Now, students classify two- dimensionalshapes in a hierarchy based on properties. Details learned in earlier gradesneed to be used in the descriptions of the attributes of shapes. The more waysthat students can classify and discriminate shapes, the better they canunderstand them. The shapes are not limited to quadrilaterals.Students can use graphic organizers such as flow charts or T-charts tocompare and contrast the attributes of geometric figures. Have students createa T-chart with a shape on each side. Have them list attributes of the shapes,such as number of side, number of angles, types of lines, etc. they need todetermine what’s alike or different about the two shapes to get a largerclassification for the shapes.Pose questions such as, “Why is a square always a rectangle?” and “Why is arectangle not always a square?”Resources/ToolsRectangles and Parallelograms: Students use dynamic software to examinethe properties of rectangles and parallelograms, and identify whatdistinguishes a rectangle from a more general parallelogram.http://illuminations.nctm.org/LessonDetail.aspx?ID=L350Using spatial relationships, they will examine the properties of two-and threedimensionalshapes.http://illuminations.nctm.org/LessonDetail.aspx?ID=L270:In this lesson, students classify polygons according to more than one propertyat a time. In the context of a game, students move from a simple descriptionof shapes to an analysis of how properties are related.MisconceptionsStudents think that when describing geometric shapes and placing them insubcategories, the last category is the only classification that can be used.Connections This cluster is connected to the <strong>Grade</strong> 5 Critical Area of Focus #3,Developing understanding of volume.111[PLSD]

CRITICAL AREA OF FOCUS #3Developing understanding of volume. Students recognize volume as anattribute of three-dimensional space. They understand that volume can bemeasured by finding the total number of same-size units of volume requiredto fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They selectappropriate units, strategies, and tools for solving problems that involveestimating and measuring volume. They decompose three-dimensional shapesand find volumes of right rectangular prisms by viewing them as decomposedinto layers of arrays of cubes. They measure necessary attributes of shapes inorder to determine volumes to solve real world and mathematical problems.Resources2-Dimensional FiguresLearn Alberta - Triangles - Video Tutorial -http://www.learnalberta.ca/content/me5l/html/<strong>Math</strong>5.html?launch=trueLearn Alberta - Polygons- Video Tutorial -http://www.learnalberta.ca/content/me5l/html/<strong>Math</strong>5.html?launch=trueIXL - Types of Triangles- Assessment –http://www.ixl.com/math/grade-5/types-of-trianglesIXL - Regular and Irregular Polygons- Assessment -http://www.ixl.com/math/grade-5/regular-and-irregular-polygonsScholastic Study Jams - Classify Triangles - Student Tutorial -http://studyjams.scholastic.com/studyjams/jams/math/geometry/classifytriangles.htmScholastic Study Jams - Classify Quadrilaterals - Student Tutorial -http://studyjams.scholastic.com/studyjams/jams/math/geometry/classifyquadrilaterals.htmCut the Knot - Triangle Classification - Teacher Tutorial –http://www.cut-the-knot.org/triangle/Triangles.shtml112[PLSD]

5 Min Life Videopedia - Classify Triangles Based on Sides and Angles -Video Tutorial –http://www.5min.com/Video/How-to-Classify-Triangles- Based-on-Sidesand-Angles-275614619113[PLSD]

Examples of Key Advances from <strong>Grade</strong> 4 to <strong>Grade</strong> 5• In grade 5, students will integrate decimal fractions more fully into theplace value system (5.NBT.1-4). By thinking about decimals as sumsof multiples of base-ten units, students begin to extend algorithms formultidigit operations to decimals (5.NBT.7).• Students use their understanding of fraction equivalence and their skillin generating equivalent fractions as a strategy to add and subtractfractions, including fractions with unlike denominators.• Students apply and extend their previous understanding ofmultiplication to multiply a fraction or whole number by a fraction(5.NF.4). They also learn the relationship between fractions anddivision, allowing them to divide any whole number by any nonzerowhole number and express the answer in the form of a fraction ormixed number (5.NF.3). And they apply and extend their previousunderstanding of multiplication and division to divide a unit fractionby a whole number or a whole number by a unit fraction.[1]• Students extend their grade 4 work in finding whole-number quotientsand remainders to the case of two-digit divisors (5.NBT.6).• Students continue their work in geometric measurement by workingwith volume as an attribute of solid figures and as a measurementquantity (5.MD.3-5).• Students build on their previous work with number lines to use twoperpendicular number lines to define a coordinate system (5.G.1-2).114[PLSD]

• When students meet this standard, they fully extend multiplication tofractions, making division of fractions in grade 6 (6.NS.1) a neartarget. 5.NF.4• Students work with volume as an attribute of a solid figure and as ameasurement quantity. 5.MD.5 Students also relate volume tomultiplication and addition. This work begins a progression leading tovaluable skills in geometric measurement in middle school.Examples of Opportunities for Connecting <strong>Math</strong>ematicalContent and <strong>Math</strong>ematical Practices (<strong>Grade</strong> 5)• <strong>Math</strong>ematical practices should be evident throughout mathematicsinstruction and connected to all of the content areas highlightedabove, as well as all other content areas addressed at this grade level.<strong>Math</strong>ematical tasks (short, long, scaffolded and unscaffolded) are animportant opportunity to connect content and practices. Some briefexamples of how the content of this grade might be connected to thepractices follow.• When students break divisors and dividends into sums of multiples ofbase-ten units (5.NBT.6), they are seeing and making use of structure(MP.7) and attending to precision (MP.6). Initially for most students,multidigit division problems take time and effort, so they also requireperseverance (MP.1) and looking for and expressing regularity inrepeated reasoning (MP.8).• When students explain patterns in the number of zeros of the productwhen multiplying a number by powers of 10 (5.NBT.2), they have anopportunity to look for and express regularity in repeated reasoning(MP.8). When they use these patterns in division, they are makingsense of problems (MP.1 ) and reasoning abstractly and quantitatively(MP.2).• When students show that the volume of a right rectangular prism isthe same as would be found by multiplying the side lengths (5.MD.5),they also have an opportunity to look for and express regularity inrepeated reasoning (MP.8). They attend to precision (MP.6) as theyuse correct length or volume units, and they use appropriate toolsstrategically (MP.5) as they understand or make drawings to showthese units.116[PLSD]

The 3-8 PARCC assessments will be delivered at each grade level andwill be based directly on the Common Core State StandardsThe 3-8 PARCC assessments will be delivered at each grade level and willbe based directly on the Common Core State Standards.The distributed PARCC design includes four components - two requiredsummative and two optional non-summative - to provide educators withtimely feedback to inform instruction and provide multiple measures ofstudent achievement across the school year.Summative Assessment Components:• Performance-Based Assessment (PBA) administered as close to the endof the school year as possible. The English language arts/literacy(ELA/literacy) PBA will focus on writing effectively when analyzingtext. The mathematics PBA will focus on applying skills, concepts,and understandings to solve multi-step problems requiring abstractreasoning, precision, perseverance, and strategic use of tools.• End-of-Year Assessment (EOY) administered after approximately 90%of the school year. The ELA/literacy EOY will focus on readingcomprehension. The mathematics EOY will call on students todemonstrate further conceptual understanding of the Major Contentand Additional and Supporting Content of the grade/course (asoutlined in the PARCC Model Content Frameworks), and demonstratemathematical fluency, when applicable to the grade.118[PLSD]

Non-Summative Assessment Components:• Diagnostic Assessment designed to be an indicator of student knowledgeand skills so that instruction, supports, and professional developmentcan be tailored to meet student needs.• Mid-Year Assessment (MYA) comprised of performance-based itemsand tasks, with an emphasis on hard-to-measure standards. Afterstudy, individual states may consider including the MYA as asummative component.• Speaking and Listening Assessment (ELA/literacy only) designed to bean indicator of students’ ability to communicate their own ideas, listento and comprehend the ideas of others, and to integrate and evaluateinformation from multimedia sources.The 3-8 assessments will include a range of item types, including innovativeconstructed response, extended performance tasks, and selected response (allof which will be computer based).119[PLSD]

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