New Jersey Student Learning Standards For Mathematics Grade 5
NEW JERSEYSTUDENT LEARNING STANDARDS FORMathematics Grade 5
New Jersey Student Learning Standards for MathematicsMathematics Grade 5In Grade 5, instructional time should focus on three critical areas: (1) developing fluencywith addition and subtraction of fractions, and developing understanding of themultiplication of fractions and of division of fractions in limited cases (unit fractionsdivided by whole numbers and whole numbers divided by unit fractions); (2) extendingdivision to 2-digit divisors, integrating decimal fractions into the place value system anddeveloping understanding of operations with decimals to hundredths, and developingfluency with whole number and decimal operations; and (3) developing understanding ofvolume.(1) Students apply their understanding of fractions and fraction models to represent theaddition and subtraction of fractions with unlike denominators as equivalent calculationswith like denominators. They develop fluency in calculating sums and differences offractions, and make reasonable estimates of them. Students also use the meaning offractions, of multiplication and division, and the relationship between multiplication anddivision to understand and explain why the procedures for multiplying and dividingfractions make sense. (Note: this is limited to the case of dividing unit fractions by wholenumbers and whole numbers by unit fractions.)(2) Students develop understanding of why division procedures work based on themeaning of base-ten numerals and properties of operations. They finalize fluency withmulti-digit addition, subtraction, multiplication, and division. They apply theirunderstandings of models for decimals, decimal notation, and properties of operations toadd and subtract decimals to hundredths. They develop fluency in these computations,and make reasonable estimates of their results. Students use the relationship betweendecimals and fractions, as well as the relationship between finite decimals and wholenumbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a wholenumber), to understand and explain why the procedures for multiplying and dividingfinite decimals make sense. They compute products and quotients of decimals tohundredths efficiently and accurately.(3) Students recognize volume as an attribute of three-dimensional space. Theyunderstand that volume can be measured by finding the total number of same-size unitsof volume required to fill the space without gaps or overlaps. They understand that a 1unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They selectappropriate units, strategies, and tools for solving problems that involve estimating andmeasuring volume. They decompose three-dimensional shapes and find volumes of rightrectangular prisms by viewing them as decomposed into layers of arrays of cubes. Theymeasure necessary attributes of shapes in order to determine volumes to solve realworld and mathematical problems.2
New Jersey Student Learning Standards for MathematicsGrade 5 OverviewOperations and Algebraic Thinking Write and interpret numerical expressions. Analyze patterns and relationships.Number and Operations in Base TenMathematical Practices1. Make sense of problems and persevere insolving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique thereasoning of others. Understand the place value system. Perform operations with multi-digit wholenumbers and with decimals to hundredths.4. Model with mathematics.Number and Operations—Fractions6. Attend to precision. Use equivalent fractions as a strategy to addand subtract fractions. Apply and extend previous understandingsof multiplication and division to multiply anddivide fractions.5. Use appropriate tools strategically.7. Look for and make use of structure.8. Look for and express regularity in repeatedreasoningMeasurement and Data Convert like measurement units within a given measurement system. Represent and interpret data. Geometric measurement: understand concepts of volume and relatevolume to multiplication and to addition.Geometry Graph points on the coordinate plane to solve real-world andmathematical problems. Classify two-dimensional figures into categories based on their properties.3
New Jersey Student Learning Standards for MathematicsOperations and Algebraic Thinking5.OAA. Write and interpret numerical expressions.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressionswith these symbols.2. Write simple expressions that record calculations with numbers, and interpret numericalexpressions without evaluating them. For example, express the calculation “add 8 and 7, thenmultiply 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.B. Analyze patterns and relationships.3. Generate two numerical patterns using two given rules. Identify apparent relationshipsbetween corresponding terms. Form ordered pairs consisting of corresponding terms fromthe two patterns, and graph the ordered pairs on a coordinate plane. For example, given therule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number0, generate terms in the resulting sequences, and observe that the terms in one sequence aretwice the corresponding terms in the other sequence. Explain informally why this is so.Number and Operations in Base Ten5.NBTA. Understand the place value system.1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as itrepresents in the place to its right and 1/10 of what it represents in the place to its left.2. Explain patterns in the number of zeros of the product when multiplying a number bypowers of 10, and explain patterns in the placement of the decimal point when a decimal ismultiplied or divided by a power of 10. Use whole-number exponents to denote powers of10.3. Read, write, and compare decimals to thousandths.a. Read and write decimals to thousandths using base-ten numerals, number names, andexpanded 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 each place, using , , and symbols to record the results of comparisons.4. Use place value understanding to round decimals to any place.B. Perform operations with multi-digit whole numbers and with decimals to hundredths.5. Fluently multiply multi-digit whole numbers using the standard algorithm.6. Find whole-number quotients of whole numbers with up to four-digit dividends and twodigit divisors, using strategies based on place value, the properties of operations, and/or therelationship between multiplication and division. Illustrate and explain the calculation byusing equations, rectangular arrays, and/or area models.7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models ordrawings and strategies based on place value, properties of operations, and/or therelationship between addition and subtraction; relate the strategy to a written method andexplain the reasoning used.4
New Jersey Student Learning Standards for MathematicsNumber and Operations—Fractions5.NFA. Use equivalent fractions as a strategy to add and subtract fractions.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacinggiven fractions with equivalent fractions in such a way as to produce an equivalent sum ordifference of fractions with like denominators. For example, 2/3 5/4 8/12 15/12 23/12. (In general, a/b c/d (ad bc)/bd.)2. Solve word problems involving addition and subtraction of fractions referring to the samewhole, including cases of unlike denominators, e.g., by using visual fraction models orequations to represent the problem. Use benchmark fractions and number sense of fractionsto estimate mentally and assess the reasonableness of answers. For example, recognize anincorrect result 2/5 1/2 3/7, by observing that 3/7 1/2.B. Apply and extend previous understandings of multiplication and division to multiply anddivide fractions.3. Interpret a fraction as division of the numerator by the denominator (a/b a b). Solveword problems involving division of whole numbers leading to answers in the form offractions or mixed numbers, e.g., by using visual fraction models or equations to representthe 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 4 people eachperson has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally byweight, how many pounds of rice should each person get? Between what two whole numbersdoes your answer lie?4. Apply and extend previous understandings of multiplication to multiply a fraction or wholenumber by a fraction.a. Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently,as the result of a sequence of operations a q b. For example, use a visual fraction modelto show (2/3) 4 8/3, and create a story context for this equation. Do the same with (2/3) (4/5) 8/15. (In general, (a/b) (c/d) ac/bd.)b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of theappropriate unit fraction side lengths, and show that the area is the same as would befound by multiplying the side lengths. Multiply fractional side lengths to find areas ofrectangles, and represent fraction products as rectangular areas.5. Interpret multiplication as scaling (resizing), by:a. Comparing the size of a product to the size of one factor on the basis of the size of theother factor, without performing the indicated multiplication.b. Explaining why multiplying a given number by a fraction greater than 1 results in a productgreater than the given number (recognizing multiplication by whole numbers greater than1 as a familiar case); explaining why multiplying a given number by a fraction less than 1results in a product smaller than the given number; and relating the principle of fractionequivalence a/b (n a)/(n b) to the effect of multiplying a/b by 1.6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., byusing visual fraction models or equations to represent the problem.5
New Jersey Student Learning Standards for Mathematics7. Apply and extend previous understandings of division to divide unit fractions by wholenumbers and whole numbers by unit fractions.1a. Interpret division of a unit fraction by a non-zero whole number, and compute suchquotients. For example, create a story context for (1/3) 4, and use a visual fraction modelto show the quotient. Use the relationship between multiplication and division to explainthat (1/3) 4 1/12 because (1/12) 4 1/3.b. Interpret division of a whole number by a unit fraction, and compute such quotients. Forexample, create a story context for 4 (1/5), and use a visual fraction model to show thequotient. Use the relationship between multiplication 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-zero whole numbersand division of whole numbers by unit fractions, e.g., by using visual fraction models andequations to represent the problem. For example, how much chocolate will each person getif 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups ofraisins?Measurement and Data5.MDA. Convert like measurement units within a given measurement system.1. Convert among different-sized standard measurement units within a given measurementsystem (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, realworld problems.B. Represent and interpret data.2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).Use operations on fractions for this grade to solve problems involving information presentedin line plots. For example, given different measurements of liquid in identical beakers, find theamount of liquid each beaker would contain if the total amount in all the beakers wereredistributed equally.C. Geometric measurement: understand concepts of volume and relate volume tomultiplication and to addition.3. Recognize volume as an attribute of solid figures and understand concepts of volumemeasurement.a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” ofvolume, and can be used to measure volume.b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said tohave a volume of n cubic units.4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and non-standardunits.5. Relate volume to the operations of multiplication and addition and solve real world andmathematical problems involving volume.a. Find the volume of a right rectangular prism with whole-number side lengths by packing itwith unit cubes, and show that the volume is the same as would be found by multiplyingthe edge lengths, equivalently by multiplying the height by the area of the base. Representthreefold whole-number products as volumes, e.g., to represent the associative property ofmultiplication.Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about therelationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.16
New Jersey Student Learning Standards for Mathematicsb. Apply the formulas V l w h and V B h for rectangular prisms to find volumes ofright rectangular prisms with whole number edge lengths in the context of solving realworld and mathematical problems.c. Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts,applying this technique to solve real world problems.Geometry5.GA. Graph points on the coordinate plane to solve real-world and mathematical problems.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, withthe intersection of the lines (the origin) arranged to coincide with the 0 on each line and agiven point in the plane located by using an ordered pair of numbers, called its coordinates.Understand that the first number indicates how far to travel from the origin in the directionof one axis, and the second number indicates how far to travel in the direction of the secondaxis, with the convention that the names of the two axes and the coordinates correspond(e.g., x-axis and x-coordinate, y-axis and y-coordinate).2. Represent real world and mathematical problems by graphing points in the first quadrant ofthe coordinate plane, and interpret coordinate values of points in the context of thesituation.B. Classify two-dimensional figures into categories based on their properties.3. Understand that attributes belonging to a category of two-dimensional figures also belong toall subcategories of that category. For example, all rectangles have four right angles andsquares are rectangles, so all squares have four right angles.4. Classify two-dimensional figures in a hierarchy based on properties.7
New Jersey Student Learning Standards for MathematicsMathematics Standards for Mathematical PracticeThe Standards for Mathematical Practice describe varieties of expertise that mathematicseducators at all levels should seek to develop in their students. These practices rest on important“processes and proficiencies” with longstanding importance in mathematics education. The firstof these are the NCTM process standards of problem solving, reasoning and proof,communication, representation, and connections. The second are the strands of mathematicalproficiency specified in the National Research Council’s report Adding 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), and productive disposition (habitual inclination to see mathematicsas sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).1 Make sense of problems and persevere in solving them.Mathematically proficient students start by explaining to themselves the meaning of a problemand looking for entry points to its solution. They analyze givens, constraints, relationships, andgoals. They make conjectures about the form and meaning of the solution and plan a solutionpathway rather than simply jumping into a solution attempt. They consider analogous problems,and try special cases and simpler forms of the original problem in order to gain insight into itssolution. They monitor and evaluate their progress and change course if necessary. Olderstudents might, depending on the context of the problem, transform algebraic expressions orchange the viewing window on their graphing calculator to get the information they need.Mathematically proficient students can explain correspondences between equations, verbaldescriptions, tables, and graphs or draw diagrams of important features and relationships, graphdata, and search for regularity or trends. Younger students might rely on using concrete objectsor pictures to help conceptualize and solve a problem. Mathematically proficient students checktheir answers to problems using a different method, and they continually ask themselves, “Doesthis make sense?” They can understand the approaches of others to solving complex problemsand identify correspondences between different approaches.2 Reason abstractly and quantitatively.Mathematically proficient students make sense of quantities and their relationships in problemsituations. They bring two complementary abilities to bear on problems involving quantitativerelationships: the ability to decontextualize—to abstract a given situation and represent itsymbolically and manipulate the representing symbols as if they have a life of their own, withoutnecessarily attending to their referents—and the ability to contextualize, to pause as neededduring the manipulation process in order to probe into the referents for the symbols involved.Quantitative reasoning entails habits of creating a coherent representation of the problem athand; considering the units involved; attending to the meaning of quantities, not just how tocompute them; and knowing and flexibly using different properties of operations and objects.3 Construct viable arguments and critique the reasoning of others.Mathematically proficient students understand and use stated assumptions, definitions, andpreviously established results in constructing arguments. They make conjectures and build alogical progression of statements to explore the truth of their conjectures. They are able toanalyze situations by breaking them into cases, and can recognize and use counterexamples.They justify their conclusions, communicate them to others, and respond to the arguments ofothers. They reason inductively about data, making plausible arguments that take into account8
New Jersey Student Learning Standards for Mathematicsthe context from which the data arose. Mathematically proficient students are also able tocompare the effectiveness of two plausible arguments, distinguish correct logic or reasoningfrom that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementarystudents can construct arguments using concrete referents such as objects, drawings, diagrams,and actions. Such arguments can make sense and be correct, even though they are notgeneralized or made formal until later grades. Later, students learn to determine domains towhich an argument applies. Student
New Jersey Student Learning Standards for Mathematics 5 Number and Operations—Fractions 5.NF. A. Use equivalent fractions as a strategy to add and subtract fractions. 1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent .
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