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Examples
can be decomposed as 
or as 
.
Clarifications
Clarification 1: Denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16 and 100.Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Expression
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
The purpose of this benchmark is to build students’ understanding from grade 3 that each fraction is composed of the sum of its unit fractions. Decomposing fractions becomes the foundation for students to make sense of adding and subtracting fractions, much like decomposing whole numbers provided the foundation for adding and subtracting whole numbers in the primary grades.- During instruction, students should show multiple ways to decompose a fraction into equivalent addition expressions with the support of models (e.g., objects, drawings, and equations).
Common Misconceptions or Errors
- Students may have difficulty decomposing mixed numbers and fractions greater than one because of misunderstanding of flexible fraction representations (e.g., is equivalent to 1). It is helpful when students’ expressions are accompanied by a model that justifies them.
Strategies to Support Tiered Instruction
- Instruction includes fraction tiles or fraction kits to physically place and see equivalent fractions of a model.
- Example:
- The teacher provides instruction that models how fractions can be decomposed in multiple ways.
- For example, using the same fraction tiles as above, students decompose multiple ways with the understanding that the value doesn't change: = + + + or = + + or = + .
- For example, using fraction circles, students combine 4 one-quarter circles and then see that there are 4 pieces that make up the whole circle. Equations are accompanied by a model that justifies them.
Instructional Tasks
Instructional Task 1 (MTR.2.1)
- Part A. Use a visual fraction model to show one way to decompose . Make sure to label each fraction part in the model, and write an equation to show how you decomposed .
- Part B. Show how you could decompose in a different way using a visual fraction model. Again, make sure to label each fraction part in the model, and write an equation to show how you decomposed .
Instructional Items
Instructional Item 1
- Which sums show ways to express ?
Related Courses
Related Access Points
Related Resources
Formative Assessment
Image/Photograph
Lesson Plans
Original Student Tutorial
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
Virtual Manipulative
MFAS Formative Assessments
Students are asked to use a visual fraction model to decompose three-fifths in two different ways.
Original Student Tutorials Mathematics - Grades K-5
Learn how to decompose a fraction into a sum of fractions with common denominators with this interactive tutorial.
Student Resources
Original Student Tutorial
Learn how to decompose a fraction into a sum of fractions with common denominators with this interactive tutorial.
Type: Original Student Tutorial
Problem-Solving Tasks
This task is a straightforward task related to adding fractions with the same denominator. The main purpose is to emphasize that there are many ways to decompose a fraction as a sum of fractions.
Type: Problem-Solving Task
The purpose of this task is for students to show they understand the connection between fraction and decimal notation by writing the same numbers both ways. Comparing and contrasting the two solutions shown below shows why decimal notation can be confusing. The first solution shows the briefest way to represent each number, and the second solution makes all the zeros explicit.
Type: Problem-Solving Task
The purpose of this task is to help students understand and articulate the reasons for the steps in the usual algorithm for converting a mixed number into an equivalent fraction. Step two shows that the algorithm is merely a shortcut for finding a common denominator between two fractions. This concept is an important precursor to adding mixed numbers and fractions with like denominators and as such, step two should be a point of emphasis. This task is appropriate for either instruction or formative assessment.
Type: Problem-Solving Task
Virtual Manipulative
This virtual manipulative allows individual students to work with fraction relationships. (There is also a link to a two-player version.)
Type: Virtual Manipulative
Parent Resources
Image/Photograph
Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.
Type: Image/Photograph
Problem-Solving Tasks
This task is a straightforward task related to adding fractions with the same denominator. The main purpose is to emphasize that there are many ways to decompose a fraction as a sum of fractions.
Type: Problem-Solving Task
The purpose of this task is for students to show they understand the connection between fraction and decimal notation by writing the same numbers both ways. Comparing and contrasting the two solutions shown below shows why decimal notation can be confusing. The first solution shows the briefest way to represent each number, and the second solution makes all the zeros explicit.
Type: Problem-Solving Task
The purpose of this task is to help students understand and articulate the reasons for the steps in the usual algorithm for converting a mixed number into an equivalent fraction. Step two shows that the algorithm is merely a shortcut for finding a common denominator between two fractions. This concept is an important precursor to adding mixed numbers and fractions with like denominators and as such, step two should be a point of emphasis. This task is appropriate for either instruction or formative assessment.
Type: Problem-Solving Task
