# Mathematics Grade 5 Year in Detail (SAMPLE)

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1 Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Whole number operations Place value with decimals Add and Subtract Decimals Add and Subtract Fractions Multiply and Divide Decimals Multiplying Fractions Dividing Fractions Volume Classifying 2-D Figures Coordinate Plane 25 days 20 days 10 days 25 days 20 days 20 days 10 days 10 days 10 days 10 days 5.NBT.A.1 5.NBT.A.1 5.NBT.B.7 5.NF.A.1 5.NBT.A.2 5.NF.B.4 5.NF.B.3 5.MD.C.3 5.G.B.3 5.G.A.1 5.NBT.A.2 5.NBT.A.3 5.MD.A.1 5.NF.A.2 5.NBT.B.7 5.NF.B.5 5.NF.B.7 5.MD.C.4 5.G.B.4 5.G.A.2 5.NBT.B.5 5.NBT.A.4 5.OA.A.1 5.MD.A.1 5.OA.A.1 5.NF.B.6 MP.1 5.MD.C.5 MP.3 5.OA.B.3 5.NBT.B.6 5.MD.A.1 MP.2 5.MD.B.2 MP.2 MP.1 MP.2 MP.4 MP.7 MP.4 5.MD.A.1 MP.7 MP.3 5.OA.A.1 MP.3 MP.2 MP.4 MP.6 4.MD.C.5 MP.6 5.OA.A.1 MP.8 MP.6 MP.1 MP.6 MP.4 MP.6 MP.7 4.MD.C.6 4.OA.C.5 5.OA.A.2 4.NBT.A.2 MP.7 MP.3 MP.7 MP.6 MP.8 4.MD.C.7 MP.2 4.NBT.A.3 4.MD.A.2 MP.4 MP.8 4.NF.B.4 4.MD.A.2 4.G.A.1 MP.6 MP.6 4.MD.A.2 4.G.A.2 MP.7 4.NF.A.1 4.G.A.3 4.OA.A.1 4.OA.A.2 4.NF.C.5 4.MD.A.2 4.OA.A.3 4.NBT.B.4 4.NBT.B.5 4.NBT.B.6 4.MD.A.2 Major Clusters Supporting Clusters Additional Clusters Other MD Measurement and Data (1, 2) OA Operations and Algebraic Thinking (1, 2, 3) G Geometry (1, 2, 3, 4) NBT Number and Operations in Base Ten (1, 2, 3, 4, 5, 6, 7) NF Number and Operations Fractions (1, 2, 3, 4, 5, 6, 7) MD Measurement and Data (3, 4, 5) MP Standards for Mathematical Practice Potential Gaps in Student Pre-Requisite Knowledge 4.OA 1, 2, 3, 5 4.NBT 2, 3, 4, 5, 6 4.NF 1, 4, 5 4.MD 5, 6, 7 4.G - 1, 2, 3 Page 1

2 This plan is meant to support districts creating their own curriculum or pacing guides. The scope and sequence of curricular resources such as Eureka Math and others will likely not match this sample plan exactly. The standards do not require only one order to achieve mastery. Thus, many curricular tools will suggest different scope and sequences for the standards. Districts should follow the guidance they feel is most appropriate for their students. Summary of Year for Grade 5 Mathematics The critical areas of this grade should be focused on three areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume. Mathematical Practices Recommendations for Grade 5 Throughout Grade 5, students should continue to develop proficiency with the Common Core s eight Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. Look for and express regularity in repeated reasoning. These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in those units. Fluency Requirements for Grade 5 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. Page 2

4 as , without having to calculate the indicated sum or product. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision. MP.7 Look for and make use of structure. As students solve problems involving measurement conversions, they will need to be able to reason about whether converting among units will result in a larger or smaller number than they began with and attend to precision as they work to find the answer (MP.2 and MP.6). Students will also make use of structure as they evaluate and write expressions involving grouping symbols (MP.7). Page 4

5 Unit 2: Place Value with Decimals 15 days Students extend their understanding of the base-ten system to decimals to the thousandths place, building on their grade 4 work with tenths and hundredths. Students use base-ten blocks and pictures of base-ten blocks to investigate the relationship between adjacent places, how numbers compare, and how numbers round to thousandths. They use their understanding of unit fractions to compare decimal places and fractional language to describe those comparisons. Students express their understanding that in multidigit whole numbers, a digit in one place represents 10 times what it represents in the place to its right and 1/10 of what it represents in the place to its left. Major Cluster Standards Understand the place value system. 5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 4.NBT.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.NBT.A.3. 5.NBT.A.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., = (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.NBT.A.3 Click to see wording of a potential gap whose content is a pre-requisite for 5.NBT.A.4. 5.NBT.A.4 Use place value understanding to round decimals to any place. Supporting Cluster Standards Convert like measurement units within a given measurement system. 4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.MD.A.1. 5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g. convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. 5.NBT.A.1 Work should include decimals and whole numbers. 5.MD.A. 1 Students should use the centimeter and/or meter stick to convert within the same system as it relates to decimals. 4.MD.A.1 include application word problems involving distances, volumes, masses, and money with decimal values Students use their understanding of structure of whole numbers to generalize the understanding to decimals (MP.7). As students continue to work with measurement conversions, they begin to generalize the procedures used (MP.8). Page 5

8 MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. Students make sense of the problems by using visual models and equations to solve problems involving the addition and subtraction of fractions (MP.1 and MP.4). Students will attend to precision MP.4 Model with mathematics. as they communicate their reasoning in the problem-solving process (MP.3 and MP.6). MP.6 Attend to precision. Page 8

9 Unit 5: Multiply and Divide Decimals 20 days Students will study the effects of multiplying and dividing by powers of 10 and be able to explain why the product is 10 or 100 times the multiplicand or the quotient is 0.1 or 0.01of the dividend. Students will compute products and quotients of decimals to hundredths efficiently and accurately. Major Cluster Standards Understand the place value system. 5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplies or divided by a power of 10. Use whole-number exponents to denote powers of NBT.B.7 Divisors could be less than one. Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Additional Cluster Standards Write and interpret numerical expressions. 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. As students work with the operations with decimals, they will reason about the relationship between operations with whole numbers and operations with decimals (MP.2). Students will explain their reasoning to the method used when performing operations with decimals using precise language (MP.3 and MP.6). Students will use structure as they evaluate expressions with grouping symbols (MP.7) and generalize the process for multiplying by powers of 10 (MP.8). Page 9

10 Unit 6: Multiplying Fractions 20 days In fourth grade, students began studying the multiplication of fractions by focusing on multiplying a whole number by a fraction. In Grade 5, students use the meaning of fractions and of multiplication to understand and explain why the procedure for multiplying fractions makes sense. Students can reason why it works using visual models, fraction strips, and number line diagrams. For more complicated problems an area model is useful, in which students work with a rectangle that has fractional side lengths, dividing it into rectangles whose side lengths are the corresponding unit fractions. They will apply the concept of multiplying fractions to find the area of a rectangle with fractional side lengths. Students learn to see products such as ½ x 3 as an expression that can be interpreted in terms of a quantity, 3, and a scaling factor, ½. They see ½ x 3 as half the size of 3 without evaluating the product. Major Cluster Standards Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 4.NF.B.4 Click to see wording of a potential gap whose content is a pre-requisite for 5.NF.B.4. 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply 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 equal parts; 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 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 the appropriate unit fraction side lengths, and how that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.B.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 the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller 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. 4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.NF.B.6. 5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision. Representing multiplication of fractions with visual and concrete models is fundamental to this unit in order for student to make sense of multiplying fractions by fractions (MP.1 and MP.4). Students reason abstractly and practice communicating their thinking in real-world situations (MP.2 and MP.6). Page 10

11 Unit 7: Dividing Fractions 10 days Students use the relationship between division and multiplication to start working with simple fraction division problems. Having seen that division of a whole number by a whole number, e.g. 5 3, is the same as multiplying the number by a unit fraction, 1/3 x 5, they now extend the same reasoning to division of a unit fraction by a whole number. Seeing for example 1/6 3 they reason that 1/6 must be divided into 3 equal parts. Since there are 6 portions of 1/6 in 1 whole and each part must be divided into 3 equal parts, there would be a total of 18 parts in the whole; therefore 1/6 3 = 1/18. In 3 1/6, they reason that since there are 6 portions of 1/6 in 1 whole, there must be 3 x 6 in 3 wholes; therefore 3 1/6 = 18. Linear and area models will be useful for student to gain a conceptual understanding of division of fractions before mastering the algorithm. Major Cluster Standards Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 5. NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. 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 numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? 5.NF.B.7 Division of a fraction by a fraction is not a requirement at this grade. MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision. In this unit it is critical for students to use concrete objects or pictures to help conceptualize, create, and solve problems (MP.1 and MP.2). Students will model the relationships in real-world problems and use precise language when explaining their reasoning (MP.4 and MP.6). Page 11

12 Unit 8: Volume 10 days Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size cube units required to 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 select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems. Major Cluster Standards Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a unit cube, is said to have one cubic unit of volume, 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 to have a volume of n cubic units. 5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 4.MD.A.2 Click to see wording of a potential gap whose content is a pre-requisite for 5.MD.C.5b. 5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. b. Apply the formulas V = l w h and V = b h for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. Students will find the volume of right rectangular prisms with edges whose lengths are whole numbers. Students learn to determine the volumes of several right rectangular prisms, using cubic centimeters, cubic inches, and cubic feet. With guidance, they learn to increasingly apply multiplicative reasoning to determine volumes, looking for and making use of structure. They understand that multiplying the length times the width of a right rectangular prism can be viewed as determining how many cubes would be in each layer if the prism were packed with or built up from unit cubes. MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. Students decompose and recompose geometric figures to make sense of the spatial structure of volume (MP.7). Students will use cubes to model volume of solid figures and connect the structure to multiplicative reasoning (MP.4 and MP.7). Students will attend to precision as they state the volume of solid figures using the appropriate units (MP.6). They solve problems by applying generalized formulas (MP.8). Page 12

13 Unit 9: Classifying 2-D Figures 10 days Students learn to analyze and relate categories of two-dimensional figures explicitly based on their properties. They classify two-dimensional figures in hierarchies. For example, they conclude that all rectangles are parallelograms, because they are all quadrilaterals with two pairs of opposite, parallel, equal-length sides. In this way, they relate certain categories of shapes as subclasses of other categories. This leads to students understanding that squares possess all properties of rhombuses and of rectangles. Additional Cluster Standards 4.MD.C.5 4.MD.C.6 4.MD.C.7 4.G.A.3 Click to see wording of potential gaps whose content is a pre-requisite for this unit. Classify two-dimensional figures into categories based on their properties. 4.G.A.1 4.GA.A.2 Click to see wording of potential gaps whose content is a pre-requisite for 5.G.B.3 and 5.G.B.4. 5.G.B.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. 5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties. MP.3 Construct viable arguments and critique the reasoning of others. Students make use of structure to build a logical progression of statements and explore hierarchical relationships among 2- MP.7 Look for and make use of dimensional shapes (MP.3 and MP.7). structure. Page 13

14 Unit 10: Coordinate Plane 10 days Students connect ordered pairs of whole-number coordinates to points on the grid, so that these coordinate pairs constitute numerical objects and ultimately can be operated upon as single mathematical entities. Students solve mathematical and real-world problems using coordinates. Students extend their Grade 4 pattern work by working briefly with two numerical patterns that can be related and examining these relationships within sequences of ordered pairs and in the graphs in the first quadrant of the coordinate plane. Additional Cluster Standards Graph points on the coordinate plane to solve real-world and mathematical problems. 5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given 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 direction of one axis, and the second number indicates how far to travel in the direction of the second axis, 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). 5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Analyze patterns and relationships. 4.OA.C.5 Click to see wording of a potential gap whose content is a pre-requisite for 5.OA.B.3. 5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. MP.4 Model with mathematics. MP.6 Attend to precision. Students will graph points in the first quadrant only. Students precisely describe the coordinates of points and the relationship of the coordinate plane to the number line (MP.6). Students both generate and identify relationships in numerical patterns, using the coordinate plan as a way of representing these relationships and patterns (MP.4). Page 14

16 Grade 5 Potential Gaps in Student Pre-Requisite Knowledge Number and Operations Fractions (NF) Go to Unit 4 Go to Unit 6 Measurement and Data (MD) Go Unit 1 Go to Unit 2 Go to Unit 3 4.NF.A.1 4.NF.B.4 4.NF.C.5 4.MD.A.2 Explain why a fraction a/b is equivalent to a fraction (n a)/(n b) 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. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 (1/4), recording the conclusion by the equation 5/4 = 5 (1/4). b. 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. For example, use a visual fraction model to express 3 (2/5) as 6 (1/5), recognizing this product as 6/5. (In general, n (a/b) = (n a)/b.) 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. For 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 what two whole numbers does your answer lie? 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. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 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. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: 5.NF.A.1 5.NF.B.4a 5.NF.A.1 5.NF.A.2 5.NF.B.6 5.MD.A.1 5.MD.C.5b Go to Unit 4 Go to Unit 6 Go to Unit 8 Go to Unit 9 4.MD.C.5 4.MD.C.6 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 one-degree angle, and can be used to measure angles. b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. Geometry Unit Page 16

17 Grade 5 Potential Gaps in Student Pre-Requisite Knowledge Measurement and Data (MD) Go to Unit 9 Geometry (G) Go to Unit 9 4.MD.C.7 4.G.A.1 4.G.A.2 4.G.A.3 Recognize angle measure as additive. When an angle is decomposed into non-overlapping 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 real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. Classify two-dimensional 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. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. Geometry Unit 5.G.B.3 5.G.B.3 5.G.B.4 Geometry Unit Page 17

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