
Mathematics (2015) 
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1.) Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. [4OA1]

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2.) Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (See Appendix A, Table 2.) [4OA2]

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3.) Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. [4OA3]


Mathematics (2015) 
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4.) Find all factor pairs for a whole number in the range 1100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1100 is prime or composite. [4OA4]


Mathematics (2015) 
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5.) Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. [4OA5]
Example: Given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence, and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.


Mathematics (2015) 
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6.) Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. [4NBT1]
Example: Recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Mathematics (2015) 
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7.) Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. [4NBT2]

Mathematics (2015) 
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8.) Use place value understanding to round multidigit whole numbers to any place. [4NBT3]


Mathematics (2015) 
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9.) Fluently add and subtract multidigit whole numbers using the standard algorithm. [4NBT4]

Mathematics (2015) 
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10.) Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. [4NBT5]

Mathematics (2015) 
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11.) Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. [4NBT6]


Mathematics (2015) 
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12.) Explain why a fraction ^{a}/_{b} is equivalent to a fraction ^{nxa}/_{nxb} by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. [4NF1]

Mathematics (2015) 
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13.) Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators or by comparing to a benchmark fraction such as ^{1}/_{2}. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. [4NF2]


Mathematics (2015) 
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14.) Understand a fraction ^{a}/_{b} with a > 1 as a sum of fractions ^{1}/_{b}. [4NF3]
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. [4NF3a]
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. [4NF3b]
Examples: ^{3}/_{8} = ^{1}/_{8} + ^{1}/_{8} + ^{1}/_{8}; ^{3}/_{8} = ^{1}/_{8} + ^{2}/_{8}; 2 ^{1}/_{8} = 1 + 1 + ^{1}/_{8} = ^{8}/_{8} + ^{8}/_{8} + ^{1}/_{8}.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. [4NF3c]
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. [4NF3d]

Mathematics (2015) 
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15.) Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. [4NF4]
a. Understand a fraction ^{a}/_{b} as a multiple of ^{1}/_{b}. [4NF4a]
Example: Use a visual fraction model to represent ^{5}/_{4} as the product 5 x (^{1}/_{4}), recording the conclusion by the equation ^{5}/_{4} = 5 x (^{1}/_{4}).
b. Understand a multiple of ^{a}/_{b} as a multiple of ^{1}/_{b}, and use this understanding to multiply a fraction by a whole number. [4NF4b]
Example: Use a visual fraction model to express 3 x (^{2}/_{5}) as 6 x (^{1}/_{5}), recognizing this product as ^{6}/_{5}. (In general, n x (^{a}/_{b}) = ^{(nxa)}/_{b}.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. [4NF4c]
Example: If each person at a party will eat ^{3}/_{8} of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed' Between which two whole numbers does your answer lie'


Mathematics (2015) 
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16.) Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.) [4NF5]
Example: Express ^{3}/_{10} as ^{30}/_{100}, and add ^{3}/_{10} + ^{4}/_{100} = ^{34}/_{100}.

Mathematics (2015) 
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17.) Use decimal notation for fractions with denominators 10 or 100. [4NF6]
Example: Rewrite 0.62 as ^{62}/_{100}; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

Mathematics (2015) 
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18.) Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. [4NF7]


Mathematics (2015) 
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19.) Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a twocolumn table. [4MD1]
Examples: Know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

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20.) Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. [4MD2]

Mathematics (2015) 
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21.) Apply the area and perimeter formulas for rectangles in realworld and mathematical problems. [4MD3]
Example: Find the width of a rectangular room given the area of the flooring and the length by viewing the area formula as a multiplication equation with an unknown factor.


Mathematics (2015) 
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22.) Make a line plot to display a data set of measurements in fractions of a unit (^{1}/_{2}, ^{1}/_{4}, ^{1}/_{8}). Solve problems involving addition and subtraction of fractions by using information presented in line plots. [4MD4]
Example: From a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.


Mathematics (2015) 
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23.) Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. [4MD5]
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through ^{1}/_{360} of a circle is called a "onedegree angle" and can be used to measure angles. [4MD5a]
b. An angle that turns through n onedegree angles is said to have an angle measure of n degrees. [4MD5b]

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24.) Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. [4MD6]

Mathematics (2015) 
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25.) Recognize angle measure as additive. When an angle is decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in realworld or mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. [4MD7]


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26.) Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. [4G1]

Mathematics (2015) 
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27.) Classify twodimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. [4G2]

Mathematics (2015) 
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28.) Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify linesymmetric figures and draw lines of symmetry. [4G3]
