Understanding Mixed Fractions

Part of

• Adding and Subtracting Fractions

Standards alignment

Common Core State Standards

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Alabama

• MA19.4.15 — Model and justify decompositions of fractions and explain addition and subtraction of fractions as joining or separating parts referring to the same whole.
• MA19.3.15a — Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers.
• MA19.4.15a — Decompose a fraction as a sum of unit fractions and as a sum of fractions with the same denominator in more than one way using area models, length models, and equations.
• MA19.4.15b — Add and subtract fractions and mixed numbers with like denominators using fraction equivalence, properties of operations, and the relationship between addition and subtraction.

Alaska

• 4.NF.3.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). 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.
• 3.NF.3.c — Express and model whole numbers as fractions, and recognize and construct 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.
• 4.NF.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Arizona

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 * — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Arkansas

• 4.NPV.7 — Decompose fractions, including fractions greater than one and mixed numbers, into unit fractions, using concrete models, drawings, and/or the number line. • Fractions include denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

California

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
• 4.NF.B.4.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 .

Colorado

• 4.NF.B.3.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. 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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: 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.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Connecticut

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Delaware

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Florida

• MA.4.FR.2.1 — Decompose a fraction, including mixed numbers and fractions greater than one, into a sum of fractions with the same denominator in multiple ways. Demonstrate each decomposition with objects, drawings and equations.
• MA.4.FR.1.3 — Identify and generate equivalent fractions, including fractions greater than one. Describe how the numerator and denominator are affected when the equivalent fraction is created.
• MA.3.FR.1.3 — Read and write fractions, including fractions greater than one, using standard form, numeral-word form and word form.

Georgia

• 4.NR.4.1 — Using concrete materials, drawings, and number lines, demonstrate and explain the relationship between equivalent fractions, including fractions greater than one, and explain the identity property of multiplication as it relates to equivalent fractions. Generate equivalent fractions using these relationships.
• 4.NR.4 — Solve real-life problems involving addition, subtraction, equivalence, and comparison of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100 using part-whole strategies and visual models.
• 4.NR.4.4 — Represent whole numbers and fractions as the sum of unit fractions.
• 4.NR.4.5 — Represent a fraction as a sum of fractions with the same denominator in more than one way, recording with an equation.

Hawaii

• 4.NF.B.3b — 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.
• 3.NF.A.3c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Idaho

• 4.NF.B.3.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 the conclusions by using a visual fraction model and/or verbal reasoning.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b
• 5.MP.4 — Model with mathematics.

Illinois

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Indiana

• 4.NS.2 — Model mixed numbers and improper fractions using visual fraction models such as number lines and area models. Use a visual fraction model to show the equivalency between whole numbers and whole numbers as fractions.
• 4.CA.6 — Add and subtract fractions with common denominators using visual fraction models. Decompose non-unit fractions to represent them as iterations of unit fractions. (E)

Iowa

• 4.NF.B.3.b — Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Be able to justify decompositions. For example, by using a visual fraction model. For example: 3/8 = 1/8 + 1/8 + 1/8 = 1/8 + 2/8 ; 21/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8 .
• 3.NF.A.3.c — 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.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b. For example, 3/4 = 1/4 + 1/4 + 1/4$.

Kansas

• 4.NF.3.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. 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.
• 3.NF.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: 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.
• 4.NF.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Kentucky

• KY.3.NF.3.c — Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers.
• KY.4.NF.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
• KY.4.NF.3.b — Decomposing a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions.
• KY.4.NF.3.c — Add and subtract mixed numbers with like denominators.

Louisiana

• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b. (Denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100.)
• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Maine

• 4.NF.B.3b — 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 to build fractions from unit fractions. 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.
• 3.NF.A.3c — Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers. Examples: 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.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Maryland

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Massachusetts

• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
• 4.NF.B.3.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 drawings or visual fraction models.
• 3.NF.A.3.c — 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.

Michigan

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Minnesota

• 5.1.2.4 — Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts. For example: When comparing 1.5 and 19/12 , note that 1.5 = 1 1/2 = 1 6/12 = 18/12, so 1.5 < 19/12.
• 4.1.2.1 — Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.
• 6.1.1.7 — Convert between equivalent representations of positive rational numbers. For example: Express 10/7 as (7+3)/7 = 7/7 + 3/7 = 1 3/7.

Mississippi

• 4.NF.3.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 (including, but not limited to: concrete models, illustrations, tape diagram, number line, area model, etc.). 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.
• 3.NF.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: 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.
• 4.NF.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Missouri

• 4.NF.B.5 — Decompose a fraction into a sum of fractions with the same denominator and record each decomposition with an equation and justification.
• 4.NF.A.2 — Recognize and generate equivalent fractions.
• 4.NF.B.4 — Understand addition and subtraction of fractions as joining/composing and separating/decomposing parts referring to the same whole.

Montana

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Nebraska

• 4.N.2.a — Explain and demonstrate how a mixed number is equivalent to a fraction greater than one and how a fraction greater than one is equivalent to a mixed number using visual fraction models and reasoning strategies.
• 3.N.2.e — Justify whole numbers as fractions and identify fractions that are equivalent to whole numbers.
• 4.N.2 — Fractions and Decimals: Students will extend understanding of fractions by equivalence and ordering and will develop an understanding of decimals.
• 4.N.3.a — Decompose a fraction into a sum of fractions with the same denominator in more than one way and record each decomposition with an equation and a visual representation.

Nevada

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

New Hampshire

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

New Jersey

• 4.NF.B.3b — 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. 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
• 3.NF.A.3c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: 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.

New Mexico

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

New York

• NY-3.NF.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• NY-4.NF.3.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.
• NY-4.NF.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
• NY-4.MD.2.a — Solve problems involving fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.

North Carolina

• NC.4.NF.3 — Understand and justify decompositions of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100. • Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. • Decompose a fraction into a sum of unit fractions and a sum of fractions with the same denominator in more than one way using area models, length models, and equations. • Add and subtract fractions, including mixed numbers with like denominators, 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, including mixed numbers by writing equations from a visual representation of the problem.

North Dakota

• 4.NO.NF.2 — Explain and demonstrate how a mixed number is equivalent to a fraction greater than one and how a fraction greater than one is equal to a mixed number using visual fraction models and reasoning strategies (proper and improper fractions limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100).
• 3.NO.NF.4 — Recognize whole numbers as fractions and express fractions that are equivalent to whole numbers.

Ohio

• 4.NF.3.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. 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.
• 3.NF.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.Examples: 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.
• 4.NF.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Oklahoma

• 5.N.1.4 — Recognize and generate equivalent terminating decimals, fractions, mixed numbers, and fractions in various models.
• 4.N.3.1 — Represent and rename equivalent fractions using fraction models (e.g., parts of a set, area models, fraction strips, number lines).
• 4.N.3.4 — Decompose a fraction into a sum of fractions with the same denominator in more than one way, using concrete and pictorial models and recording results with numerical representations (e.g., 3/4 = 1/4 + 1/4 + 1/4 and 3/4 = 2/4 + 1/4).

Oregon

• 4.NF.B.3 — Understand a fraction (a/b) as the sum (a) of fractions of the same denominator (1/b). Solve problems in authentic contexts involving addition and subtraction of fractions referring to the same whole and having like denominators.
• 4.NF.A.1 — Use visual fraction representations to recognize, generate, and explain relationships between equivalent fractions.
• 3.NF.A.3 — Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
• MP8 — Look for and express regularity in repeated reasoning

Pennsylvania

• CC.2.1.4.C.2 — Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
• CC.2.1.4.C.1 — Extend the understanding of fractions to show equivalence and ordering.

Rhode Island

• 4.NF.B.3.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 drawings or visual fraction models. 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.
• 3.NF.A.3.c — 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.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
• 4.MD.A.2 — 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.

South Carolina

• 3.NSF.3 — Develop an understanding of mixed numbers (i.e., denominators 2, 3, 4, 6, 8, 10) as iterations of unit fractions on a number line.
• 4.NSF.3 — Develop an understanding of addition and subtraction of fractions (i.e., denominators 2, 3, 4, 5, 6, 8, 10, 12, 25, 100) based on unit fractions.
• 4.PS.7a — Recognize complex mathematical objects as being composed of more than one simple object.

South Dakota

• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3.b — Decompose a fraction into a sum of fractions with like denominators in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b. For example, 4/5 = 1/5 + 1/5 + 1/5 + 1/5.
• 4.NF.B.3.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.

Tennessee

• 4.NF.B.3.b — Decompose a fraction into a sum of fractions with the same denominator in more than one way (e.g., 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) recording each decomposition by an equation. Justify decompositions using a visual fraction model.
• 3.NF.A.3.c — 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 on a number line diagram.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b. For example, 4/5 = 1/5 +1/5 + 1/5 + 1/5.

Texas

• 4.3A — represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b>0, including when a>b;
• 4.3B — decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations;
• 4.3C — determine if two given fractions are equivalent using a variety of methods;

Utah

• 4.NF.3.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, for example, by using a visual fraction model. For example, 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8; 2 1/8 = 8/8 + 8/8 + 1/8.
• 3.NF.3.c — 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.

Vermont

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Virginia

• 3.NS.3c — Use a model of a fraction greater than one to count the fractional parts to name and write it as an improper fraction and as a mixed number (e.g., 1/4, 2/4, 3/4, 4/4, 5/4 = 1 1/4 ).
• 4.NS.3f — Compose and decompose fractions (proper and improper) and/or mixed numbers with denominators of 12 or less, in multiple ways, with and without models.*
• 3.NS.3d — Compose and decompose fractions (proper and improper) with denominators of 2, 3, 4, 5, 6, 8, and 10 in multiple ways (e.g., 7/4 = 4/4 + 3/4 or 4/6 = 3/6 + 1/6 = 2/6 + 2/6 ) with models.

Washington

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
• 4.MD.A.2 — 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.

Washington, D.C.

• 4.NF.B.3.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.
• 3.NF.A.3.c — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

West Virginia

• M.4.14 — Understand the fraction a/b, with a> 1, as the sum of a of the fractions 1/b. a. Add and subtract fractions with like denominators. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 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 by using a visual fraction model (e.g., 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 by replacing each mixed number with an equivalent fraction greater than one and/or by using properties of operations and the relationship between addition and subtraction. Identify the two whole numbers a mixed number is between. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators by using visual fraction models and equations to represent the problem.
• M.3.17 — Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. a. Understand two fractions as equivalent ( equal) if they are the same size or the same point on a number line. b. 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). c. Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers (e.g., 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). d. 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). Instructional Note: Fractions in this standard am limited to de~nominators of 2, 3, 4, 6, and 8.
• MHM8 — Look for and express regularity in repeated reasoning.

Wisconsin

• M.4.NF.B.3 — Understand composing and decomposing fractions. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of unit fractions and/or multiples of that unit fraction in more than one way, recording each decomposition by an equation. Justify decompositions with explanations, visual fraction models, or equations. For example: 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 fractions, including mixed numbers, with like denominators (e.g., 3/8 + 2/8) and related denominators (e.g., 1/2 +1/4, 1/3 + 1/6) by using visual fraction models (e.g., tape diagrams and number lines), properties of operations, and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions with like and related denominators, including mixed numbers, by using visual fraction models and equations to represent the problem. Students are not required to rename fractions in lowest terms nor use least common denominators.
• M.3.NF.A.1 — Understand a unit fraction as the quantity formed when a whole is partitioned into equal parts and explain that a unit fraction is one of those parts (e.g., 1/4). Understand fractions are composed of unit fractions.
• M.3.NF.A.3 — Explain equivalence of fractions and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size or name the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3) and explain why the fractions are equivalent by using a visual fraction model (e.g., tape diagram or number line). c. Express whole numbers as fractions (3 = 3/1), and recognize fractions that are equivalent to whole numbers (4/4 = 1). d. 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. Justify the conclusions by using a visual fraction model (e.g., tape diagram or number line) and describe the result of the comparison using words and symbols ( >, =, and < ).

Wyoming

• 3.NF.F.3.C — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
• 4.NF.G.3 — Understand a fraction a/b with a > 1 as a sum of unit fractions (1/b).
• 4.NF.G.3.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 by using a visual fraction model.
• 4.NF.G.4.A — Understand a fraction a/b as a multiple of 1/ b.