In this unit, students begin to integrate their understanding of fractions with their understanding of operations. The unit begins with the idea of "reasoning up and down", which reinforces the meaning of fractions and the components of fractions. Students reason from nonunit fractions "down" to unit fractions, and then reason "up" to wholes and then to other fractions with the same denominator. This process prepares students for future reasoning with ratios, and ensures understanding of the numerator and denominator.
Students then use their grasp of addition and subtraction to discover the different ways of composing and decomposing fractions, which they have previously only partitioned into unit fractions. The emphasis on lengths helps students connect their understanding of types of addition and subtraction problems with whole numbers to the new type of number: fractions. In the final three lessons, operations and decomposition are further extended to include mixed numbers.
By rewriting, decomposing, adding, and subtracting fractions, students deepen their understanding of equivalence and operations. Students see equivalent lengths throughout the unit, reinforcing an idea they learned in previous grades about the invariance of a length. For instance, a length of 2/8 is determined by a point that is a distance of 2/8 from zero, but that length occurs in infinitely many places along the number line.
Key Concepts:
 Based on the definition of a fraction a/b as a parts that are the size of 1/b, we can construct a whole given any fractional part of that whole.
 Fractions can be decomposed into nonunit fractions in more than one way.
 The meanings of addition and subtraction do not change when those operations are used with fractions.
 Mixed numbers are a form of numbers greater than 1 that have a whole that can be decomposed in many different ways.
Prior Knowledge Needed:
 Fractions can be decomposed into unit fractions (Grade 3, Unit 4; 3.NF.A.1 (fractions between 0 and 1 only)).
 A mixed number is a representation of a fraction that uses a whole number and a fraction together (Grade 3, Unit 5; 3.NF.A.1 (fractions >1 included)).
Units on the Horizon:
 Fraction equivalence and comparison (Grade 4, Unit 5: 4.NF.A).
 Solving word problems involving fractions and mixed numbers (Grade 4, Unit 9, 4.NF.B).
Lessons

Lesson objective: Understand that we can construct a whole from any fraction a/b by knowing that fractions can be decomposed into unit fractions. This is based on the definition of the fraction a/b as determined by the number of 1/b sized pieces, and b ...

Lesson objective: Use the meaning of fractions to construct a whole given a fraction less than 1 or greater than 1. Fluency and Procedural Skill Lesson: This lesson helps to build fluency with reasoning about fractions. Students work to construct one fr...

Lesson objective: Understand that because the fraction a/b is determined by the number of 1/b sized pieces, we can construct any other fraction with the same denominator using iterations of the unit fraction 1/b. Fluency and Procedural Skill Lesson: Thi...

Lesson Objective: Solve a realworld problem by reasoning about fractions. Application Lesson: This lesson provides an opportunity for students to apply their knowledge and understanding that nonunit fractions are composed of unit fractions to reason a...

Lesson objective: Understand that a fraction can be composed from a combination of unit and nonunit fractions, or just from unit fractions, and that we can decompose a fraction into a sum of unit fractions or nonunit fractions. The concept is develope...

Lesson Objective: Understand that the meaning of addition and subtraction extend to fractions. This lesson helps build conceptual understanding that the operations of addition and subtraction can be used with fractions. The meaning and uses of the opera...

Lesson objective: Add fractions with like denominators using a number line. This lesson helps to build procedural skill in adding fractions with like denominators. Students work toward this goal in 3 ways: By using their understanding of decomposing fra...

Lesson objective: Practice subtracting fractions with like denominators. Procedural skills lesson: This lesson helps to build procedural skill in subtracting fractions with like denominators. Students work toward this goal in 3 ways: 1. By using their u...

Lesson objective: Solve word problems by adding and subtracting fractions with like denominators. Application lesson: This lesson provides an opportunity for students to apply their knowledge and understanding of fraction addition and subtraction to a r...

Lesson objective: Decompose mixed numbers. Conceptual Lesson: This lesson helps build conceptual understanding of a form of number students were introduced to in grade 3: mixed numbers. Earlier in the unit, students worked with fractions greater than 1,...

Lesson objective: Practice decomposing fractions and mixed numbers by writing equivalent expressions. Procedural skills lesson: This lesson helps to build procedural skill in decomposing fractions and mixed numbers and writing these numbers in equivalen...

Lesson objective: Decompose mixed numbers to solve word problems. Application Lesson: This lesson provides an opportunity for students to apply their knowledge and understanding of decomposition of mixed numbers to a reallife situation. The scenario fr...