# Number and Operations Fractions

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1 Number and Operations Fractions Standards Entry Points Access Skills 3 Page 59 Pages Pages Pages Page Pages Page MATHEMATICS 58

2 CONTENT AREA Mathematics DOMAIN Number and Operations Fractions GRADE 3 Mathematics Number and Operations Fractions Cluster Develop understandin g of fractions as numbers for fractions with denominators 2, 3, 4, 6, and 8. Standards as written 3.NF.A.1 3.NF.A.2 3.NF.A.2a 3.NF.A.2b 3.NF.A.3 Grade 3 Understand a fraction 1 b as the quantity formed by 1 part when a whole (a single unit) is partitioned into b equal ; understand a fraction a b as the quantity formed by a of size 1 b. Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a unit fraction, 1 b, on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal. Recognize that each part has size 1 b and that the fraction 1 b is located 1 b of a whole unit from 0 on the number line. Represent a fraction a b on a number line diagram by marking off a lengths 1 b from 0. Recognize that the resulting interval has size a b and that its endpoint locates the number a b on the number line. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 3.NF.A.3a 3.NF.A.3b 3.NF.A.3c 3.NF.A.3d Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Recognize and generate simple equivalent fractions, e.g., 1 2 = 2 4, 4 6 = 2 3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. For example, express 3 in the form 3 = 3 1; recognize that 6 1 = 6; locate 4 4 and 1 at the same point of a number line diagram. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. MATHEMATICS 59

3 ENTRY POINTS and ACCESS SKILLS for Number and OperationsFractions Standards in Grade 3 Develop understandin g of fractions as numbers for fractions with denominator s 2, 3, 4, 6, and 8. ACCESS SKILLS The student will: The student will: Respond to materials that demonstrate objects that can be Track materials that demonstrate that objects can be Shift focus from materials that demonstrate that objects can be Grasp materials that demonstrate that objects can be Use two hands to hold materials that demonstrate that objects can be Release materials that demonstrate that objects can be Move materials that demonstrate that objects can be Explain what the numerator and denominator of a fraction represent (e.g., in 2/4, the 4 tells you how many the whole is divided into and 2 is the number of you have and use a drawing to illustrate) Identify concepts of whole and half using manipulatives and/or familiar objects (e.g., using sets of objects or shapes with shaded, identify halves and wholes) Partition a whole into 2, 3 or 4 equal using visual models, number lines, or manipulatives (e.g., given a rectangle, draw lines to divide it into 3 equal ) Compare fractions of the same whole to determine which is greater (e.g., show a number line with points at ¼ and ¾ and ask which is greater) ENTRY POINTS The student will: The student will: Compare visual representations of fractions using the terms greater than, less than, or equal to (e.g., verbalize that ½ is greater than ¼) Match visual representations of simple fractions to the name of the fraction (e.g., given visual fraction models with sections already shaded in, identify amounts such as ¾, ½, 5/8) Compare of a whole (quarters, thirds, halves) to determine relative size of each (1/2, 1/3, 1/4) using manipulatives or visual models (e.g., use manipulatives to show that ½ > 1/3) Label unit fractions* on a number line (e.g., given a number line labeled with 0 and 1 and divided into 6 equal, plot and label a point at 1/6) Identify of a whole using visual fraction models (e.g., using shapes and having one part shaded in, identify 1/2, 1/3, 1/4, 1/6, 1/8) Divide a number line into equal and label points (e.g., mark off a number line labeled with 0 and 1 into 6 equal sections and label each point with 1/6, 2/6, 3/6, etc.) Record results of the comparisons of two fractions with like denominators or like numerators using symbols (e.g., use the symbols <, =, or > to write a comparison of 1/4 and ¾, or 2/4 and 2/8) Create visual representation of simple fractions (e.g., divide a shape into 4 equal and shade in ¾ recognizing that the do not need to be touching) MATHEMATICS 60

5 ACCESS SKILLS (continued) for Number and OperationsFractions Standards in Grade 3 Develop understandin g of fractions as numbers for fractions with denominators 2, 3, 4, 6, and 8. (continued) ACCESS SKILLS The student will: Choose from an array of two in a fraction based activity (e.g., choose materials to be ) Attend visually, aurally, or tactilely to materials that demonstrate objects that can be ENTRY POINTS The student will: MATHEMATICS 62

6 CONTENT AREA Mathematics DOMAIN Number and Operations Fractions GRADE 4 Mathematics Number and Operations Fractions Cluster Extend understanding of fraction equivalence and ordering for fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Standards as written 4.NF.A.1 4.NF.A.2 Grade 4 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 numbers and sizes of the differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions, including fractions greater than 1. 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. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers for fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and NF.B.3 Understand a fraction a b with a > 1 as a sum of fractions 1 b.. 4.NF.B.3a 4.NF.B.3b 4.NF.B.3c 4.NF.B.3d 4.NF.B.4 Understand addition and subtraction of fractions as joining and separating referring to the same whole. (The whole can be a set of objects.). 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 drawings or visual fraction models. Examples: 3 8 = ; 3 8 = ; = = 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. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using drawings or visual fraction models and equations to represent the problem. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.B.4a Understand a fraction a b as a multiple of 1 b.. 4.NF.B.4b 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 ). 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.) MATHEMATICS 63

7 Understand decimal notation for fractions, and compare decimal 4.NF.B.4c 4.NF.C.5 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 , and add = NF.C.6 Use decimal notation to represent fractions with denominators 10 or NF.C.7 For example, rewrite 0.62 as ; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 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. MATHEMATICS 64

8 ENTRY POINTS for Number and Operations Fractions Standards in Grade 4 Extend understanding of fraction equivalence and ordering for fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. The student will: The student will: The student will: Distinguish between equal and non-equal of a whole (e.g., compare unit fractions* or fractions with like denominators but unlike numerators using models or number lines) Identify a fraction 1/b as the quantity formed by one part when a whole is partitioned into b equal (e.g., given several shapes with unit fractions* shaded in, identify the of the whole as ½, ¼, 1/8, etc.) Demonstrate fractions equivalent to ½ using fraction models, manipulatives and/or technology (e.g., show that 2/4 = ½ on a number line) Compare two fractions with like denominators by comparing their relative size using fraction models (e.g., use shapes that are shaded in with 3/10 and 6/10 and know that 3/10 < 6/10) Identify which of two fractions represents a larger part of a whole using fraction models or manipulatives (e.g., given two number lines marked from 0 to 1 with points plotted for 4/5 and 3/8, determine that 4/5 is larger because it is more of the whole because it is to the right of 3/8) Identify equivalent fractions using fraction models, manipulatives, and/or technology (e.g., given several rectangles already shaded in with different fraction amounts, show that 2/6 = 1/3) Compare visual models of two fractions with like denominators using <, >, or = (e.g., 5/6 > 3/6) Determine which of two fractions with like denominators represents a larger part of a whole by representing the fractions with fraction models or manipulatives (e.g., given two congruent rectangles divided into eighths, shading in one to represent 3/8 and another to represent 7/8, and then identifying which is more of the whole) Generate multiple pairs of equivalent fractions using fraction models, manipulatives and/or technology (e.g. show that 2/4 = 3/6 by drawing two congruent rectangles with one divided into fourths and one divided into sixths and shading in 2/4, 3/6, 4/8 to show they are equivalent areas) Compare two fractions with unlike denominators by demonstrating which is greater or less than the benchmark of ½ using fraction models, manipulatives or technology (e.g., showing on two number lines that 2/8 < 3/4 because 2/8 is to the left of ½ on the number line and 3/4 is to the right of ½) Demonstrate, using fraction models or manipulatives, that the whole is equal to the sum of the partitioned (e.g., 4/4 = 1; 1 = 8/8) Compare visual models of fractions with unlike denominators using symbols (<, >, or =) (e.g., given two models, one of 4/5 and one with 3/10, determine which is greater and then write 4/5 > 3/10) Continue to address skills and concepts that approach gradelevel expectations in this cluster See entry points for earlier grades in this or a related cluster that are challenging and use age-appropriate materials MATHEMATICS 65

10 ENTRY POINTS for Number and Operations Fractions Standards in Grade 4 Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers for fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. (continued) The student will: The student will: The student will: Multiply a non-unit fraction by a whole number using visual models, manipulatives, or technology (e.g., 3 X 2/4 =2/4 + 2/4 + 2/4 also equals 3 groups of 2/4 and each 2/4 = ¼ + ¼) Solve a multiplication word problem involving multiplying a fraction by a whole number using visual models, manipulatives, or technology (e.g., Danny rode his bike 3/8 mile each day in the summer. What is the total distance Danny rode over 5 days? 5 X 3/8 = 15/8 or 1 7/8 miles) Understand decimal notation for fractions, and compare decimal Order decimals on a number line (e.g., given a number line labeled with tenths in decimals, plot points at 0.4, 0.7, and 0.8 or given a number line labeled with hundredths in decimals, plot points at 0.62, 0.64, and 0.68) See entry points for earlier grades in this or a related cluster that are challenging and use age-appropriate materials Show that a fraction with a denominator of ten is equivalent to a fraction with a denominator of 100 by using visual models, manipulatives, or technology (e.g., using base-ten blocks, show that 5/10 = 50/100) Continue to address skills and concepts that approach gradelevel expectations in this cluster Express a fraction with a denominator of ten as an equivalent fraction with a denominator of 100 (e.g., 3/10 = 30/100 or 60/100 = 6/10) Use decimal notation for fractions with denominators of ten. (e.g., 2/10=0.2) Compare two decimals to the tenths by reasoning about their size using symbols (<, >, or =) or visual model (e.g., use a number line to show that 0.65 > 0.40 because 0.40 is to the left of 0.65 and write the inequality) Continue to address skills and concepts that approach gradelevel expectations in this cluster MATHEMATICS 67

12 to multiply and divide (continued) 5.NF.B.5b 5.NF.B.6 5.NF.B.7 5.NF.B.7a 5.NF.B.7b 5.NF.B.7c 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. Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit 1 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. 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 x (1 5) = 4. 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 three people share ½ lb. of chocolate equally? How many 1 3 -cup servings are in two cups of raisins? 1 Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade. MATHEMATICS 69

14 ENTRY POINTS for Number and Operations Fractions Standards in Grade 5 Use equivalent fractions as a strategy to add and subtract (continued) The student will: The student will: The student will: Show valid comparisons of fractions that refer to the same whole (e.g., given two pictures of wholes divided into 6ths and 12ths, show that 2/6 is equal to 4/12) Show that comparisons are valid only if the fractions refer to the same whole (e.g., given pictures of pairs of fractions, some with different size wholes and some with the same size wholes, be able to choose the correct comparisons: = is not true because the wholes are not the same size) Apply and extend previous understandings of multiplication and division to multiply and divide Show/express that a unit fraction* as is represented by the division of the numerator by the denominator (e.g., show that 1/5 means to dividing one whole into 5 equal ) Write a multiplication problem involving a whole number and a fraction as a repeated addition problem (e.g., 4 X 3/5 = 3/5 + 3/5 + 3/5 + 3/5) Show that multiplying a fraction by a fraction is similar to creating a model of the first fraction, then scaling each part by the other fraction (e.g. ¼ X ½ can be modeled by describing or drawing a whole rectangle divided into 4 equal, and then each of the 4 (1/4) is divided into 2 equal so that each part is now 1/8 of the whole rectangle) Match fractions with their equivalent division expressions (e.g. 3/8 = 3 8, not 8 3 nor 3 X 8) Represent connections between fractions and division with the use of visual models, manipulatives or technology (e.g., show 8/4 can be represented as 8 candy bars into 4 groups results in each group getting 2 candy bars) Multiply a whole number by a unit fraction* using a number line marked from 0 to 1(e.g. use a number line labeled with ¼ s to find 3 X ¼ = ¾) Compare products of fractions and whole numbers based on the multiple using a visual fraction model (e.g., is 5 X 2/5 greater or less than 5 X 4/5 or is 3/5 X 1/2 greater or less than 7/6 X 1/2) Continue to address skills and concepts that approach gradelevel expectations in this cluster Identify division in a realworld problem as a fraction (e.g., write 5 cookies divided equally by 10 people means each person gets 5/10 or ½ of a cookie ) Multiply fractions by fractions using manipulatives, visual models and/or technology (e.g., 2/4 X 4/5 = 8/20) Multiply a whole number by a fraction less than 1 using visual models or manipulatives (e.g., show 2/3 X 4 = 8/3 by using 4 shapes that are determined to be 2/3 of a whole or by defining a whole and using 2 one-third shapes as the 2/3 and compiling 4 sets of the 2/3) MATHEMATICS 71

15 ENTRY POINTS for Number and Operations Fractions Standards in Grade 5 Apply and extend previous understandings of multiplication and division to multiply and divide (continued) The student will: The student will: The student will: Solve real-world problems involving division of a whole into equal (e.g., divide a whole candy bar to share with four friends) See entry points for earlier grades in this or a related cluster that are challenging and use age-appropriate materials Show that dividing a fraction by a whole number will give a quotient that is less than the original using fraction models or manipulatives (e.g. ½ 3 will be equal to a fraction that is less than ½) Show that dividing a whole number by a fraction will give a quotient that is greater than the original whole number using fraction models or manipulatives (e.g., 3 ½ will be greater than 3) Unit Fraction: a fraction with a numerator of one Compare the size of a product to the size of one factor (with one factor being a fraction and one factor a whole number) without performing the multiplication (e.g., know that 2/3 X 4 < 4 because 2/3 < 1 so the product must be less than 1 X 4) Solve real-world problems by multiplying fractions or mixed numbers using manipulatives, visual models, and/or technology (e.g., solve There are 4 students who need paint for their art project. Each student needs 1¾ gallons of paint. What is the total amount of paint, in gallons, needed by these four students? by using a visual fraction model to show 1¾ four times to find solution) Solve real-world problems by dividing a whole number by a unit fraction* or a unit fraction by a whole number (e.g., 3 pizzas are each divided into fourths so there are 12 pieces, 3 1/4 = 12) Connect division and multiplication of fractions (e.g., know that 5 ½ = 10 because 10 X ½ = 5) Continue to address skills and concepts that approach gradelevel expectations in this cluster MATHEMATICS 72

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