Word (.docx)
Acrobat (.pdf)
Examples
The sum of
and 
can be determined as 
,
, 
or 
by using different common denominators or equivalent fractions.
Clarifications
Clarification 1: Instruction includes the use of estimation, manipulatives, drawings or the properties of operations.Clarification 2: Instruction builds on the understanding from previous grades of factors up to 12 and their multiples.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- NA
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
The purpose of this benchmark is for students to understand that when adding or subtracting fractions with unlike denominators, equivalent fractions are generated to rewrite the fractions with like denominators, with which students have experience from grade 4 (MA.4.FR.2.2). Procedural fluency will be achieved in grade 6 (MA.6.NSO.2.3).- During instruction, have students begin with expressions with two fractions that require the rewriting of one of the fractions (where one denominator is a multiple of the other, like 1 + 3 or + ) and progress to expressions where both fractions must be rewritten (where denominators are not multiples of one another, like + or 1 + 9 ). In doing so, students can explore how both fractions need like denominators to make addition and subtraction easier. Once students have stronger conceptual understanding, expressions requiring adding or subtracting 3 or more numbers should be included in instruction.
- It is important for students to practice problems that include various fraction models as students may find that a circular model might not be the best model when adding or subtracting fractions because of the difficulty in partitioning the pieces so they are equal (MTR.2.1).
- When students use an algorithm to add or subtract fractions, encourage students’ use of
flexible strategies.
- For example, students can use a partial sums strategy when adding 1 + 4 by adding the whole numbers 1 + 4 together first before adding the fractional parts and regrouping when necessary.
- Mental computations and estimation strategies should be used to determine the
reasonableness of solutions.
- For example, when adding 1 + 4 , students could reason that the sum will be greater than 6 because the sum of the whole numbers is 5 and the sum of the fractional parts in the mixed numbers will be greater than 1. Keep in mind that estimation is about getting reasonable solutions and not about getting exact solutions, therefore allow for flexible estimation strategies and expect students to justify them.
- Although not required, instruction may include students using equivalent fractions to simplify answers.
Common Misconceptions or Errors
- Students can carry misconceptions from grade 4 about adding and subtracting fractions and understanding why the denominator remains the same. Emphasize the use of area and number line models, and present expressions in numeral-word form to help understand that the denominator is the unit.
- For example, “5 eighths + 9 eighths is equal to how many eighths?”
- Students often try to use different models when adding, subtracting or comparing
fractions.
- For example, they may use a circle for thirds and a rectangle for fourths, when comparing fractions with thirds and fourths.
- Remind students that the representations need to be from the same whole models with the
same shape and same size. In a real-world problem, this often looks like same units.
- For example, “Trey has 1 cups of water and Rachel has 2 cups of water. How many cups of water do they have?”
Strategies to Support Tiered Instruction
- Instruction includes concrete models and drawings that help solidify understanding that when adding and subtracting with unlike denominators, the value of the fractional parts remains the same.
- For example, students create a model for each of the fractions in the problem − .
- It is important for students to draw these two models the same size. Once the models are created, students will then need to be able to make all the pieces within each model the same size to be able to subtract. They then divide each piece of the model into fourths. They then divide each piece of the model into thirds. Now both models are divided in to 12 pieces and the subtraction problem can be represented as − . It is important to note that the area of the models did not change. Just because the fraction changed, the value of the fraction did not change.
- Instruction includes concrete models and drawings that help solidify understanding that when adding and subtracting with unlike denominators, students are adding and subtracting pieces of the whole.
- For example, the teacher emphasizes the use of area and number line models and presents expressions in numeral-word form to help understand that one over the denominator is the unit.
- For example, “3 twelfths + 6 twelfths are equal to how many twelfths?” The denominator is 12 so one unit is equal to 1 twelfth.
Instructional Tasks
Instructional Task 1 (MTR.2.1)
Write an expression for the visual model below. Then find the sum.
Instructional Task 2 (MTR.2.1)
Use a visual fraction model to find the value of the expression + .
Instructional Task 3 (MTR.3.1)
Find the value of the expression 3 + .
Instructional Task 4 (MTR.3.1)
Find the differences − and 2 − .
Instructional Items
Instructional Item 1
Find the sum + .- a. 1
- b.
- c. 1
- d.
Instructional Item 2
- a. 1
- b. 1
- c. 1
- d. 2
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive
Related Courses
Related Access Points
Related Resources
Educational Game
Formative Assessments
Lesson Plans
Original Student Tutorial
Perspectives Video: Expert
Perspectives Video: Teaching Idea
Problem-Solving Tasks
Student Center Activity
Teaching Idea
Tutorials
Virtual Manipulative
MFAS Formative Assessments
Students are asked to add two pairs of fractions with unlike denominators.
Students are asked to add pairs of fractions with unlike denominators.
Students are asked to subtract improper fractions and mixed numbers with unlike denominators.
Original Student Tutorials Mathematics - Grades K-5
Explore how to add fractions less than one with unlike denominators in this magical, interactive tutorial.
Student Resources
Original Student Tutorial
Explore how to add fractions less than one with unlike denominators in this magical, interactive tutorial.
Type: Original Student Tutorial
Educational Game
Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit.
Type: Educational Game
Problem-Solving Tasks
The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum so this may be given to students who are limited to computing sums of fractions with the same denominator. Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator) and reason about their relative size.
Type: Problem-Solving Task
The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting. The two primary ways one can expect students to add are converting the mixed numbers to fractions greater than 1 or adding the whole numbers and fractional parts separately. It is good for students to develop a sense of which approach would be better in a particular context.
Type: Problem-Solving Task
The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and forth between the abstract representation of the fractions and the meaning of the fractions in the context.
Type: Problem-Solving Task
The purpose of this task is to present students with a situation where it is natural to add fractions with unlike denominators; it can be used for either assessment or instructional purposes. Teachers should anticipate two types of solutions: one where students calculate the distance Alex ran to determine an answer, and one where students compare the two parts of his run to benchmark fractions.
Type: Problem-Solving Task
Part (a) of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least common denominator. Part (b) does not ask students to do it in more than one way; the purpose is to give them an opportunity to choose a denominator and possibly compare with another student who chose a different denominator. The purpose of part (c) is to help students move away from a reliance on drawing pictures. Students can draw a picture if they want, but this subtraction problem is easier to do symbolically, which helps students appreciate the power of symbolic notation.
Type: Problem-Solving Task
Part (a) of this task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common denominator to find a solution, not just the least common denominator. Part (b) does not ask students to solve the given addition problem in more than one way. Instead, the purpose of this question is to give students an opportunity to choose a denominator and possibly to compare their solution method with another student who chose a different denominator. The purpose of part (c) is to give students who are ready to work symbolically a chance to work more efficiently.
Type: Problem-Solving Task
One goal of this task is to help students develop comfort and ease with adding fractions with unlike denominators. Another goal is to help them develop fraction number sense by having students decompose fractions.
Type: Problem-Solving Task
This task addresses common errors that students make when interpreting adding fractions word problems. It is very important for students to recognize that they only add fractions when the fractions refer to the same whole, and also when the fractions of the whole being added do not overlap. This set of questions is designed to enhance a student's understanding of when it is and is not appropriate to add fractions.
Type: Problem-Solving Task
Tutorials
This tutorial explores the addition and subtraction of fractions with unlike denominators. Using the number line, this mathematical process can be easily visualized and connected to the final strategy of multiplying the denominators (a/b + c/d = ad +bc/bd). The video number line does show negative numbers which goes beyond elementary standards so an elementary teacher would need to reflect on whether this video will enrich student knowledge or cause confusion.
Type: Tutorial
In this tutorial, students will be exposed to the strategy of finding the least common denominator for certain cases. Elementary teachers should note this is not a requirement for elementary standards and consider whether this video will further student knowledge or create confusion. This chapter explains how to find the smallest possible common denominator. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.
Type: Tutorial
This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting fractions. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.
Type: Tutorial
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
Problem-Solving Tasks
The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum so this may be given to students who are limited to computing sums of fractions with the same denominator. Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator) and reason about their relative size.
Type: Problem-Solving Task
The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting. The two primary ways one can expect students to add are converting the mixed numbers to fractions greater than 1 or adding the whole numbers and fractional parts separately. It is good for students to develop a sense of which approach would be better in a particular context.
Type: Problem-Solving Task
The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and forth between the abstract representation of the fractions and the meaning of the fractions in the context.
Type: Problem-Solving Task
The purpose of this task is to present students with a situation where it is natural to add fractions with unlike denominators; it can be used for either assessment or instructional purposes. Teachers should anticipate two types of solutions: one where students calculate the distance Alex ran to determine an answer, and one where students compare the two parts of his run to benchmark fractions.
Type: Problem-Solving Task
Part (a) of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least common denominator. Part (b) does not ask students to do it in more than one way; the purpose is to give them an opportunity to choose a denominator and possibly compare with another student who chose a different denominator. The purpose of part (c) is to help students move away from a reliance on drawing pictures. Students can draw a picture if they want, but this subtraction problem is easier to do symbolically, which helps students appreciate the power of symbolic notation.
Type: Problem-Solving Task
Part (a) of this task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common denominator to find a solution, not just the least common denominator. Part (b) does not ask students to solve the given addition problem in more than one way. Instead, the purpose of this question is to give students an opportunity to choose a denominator and possibly to compare their solution method with another student who chose a different denominator. The purpose of part (c) is to give students who are ready to work symbolically a chance to work more efficiently.
Type: Problem-Solving Task
One goal of this task is to help students develop comfort and ease with adding fractions with unlike denominators. Another goal is to help them develop fraction number sense by having students decompose fractions.
Type: Problem-Solving Task
This task addresses common errors that students make when interpreting adding fractions word problems. It is very important for students to recognize that they only add fractions when the fractions refer to the same whole, and also when the fractions of the whole being added do not overlap. This set of questions is designed to enhance a student's understanding of when it is and is not appropriate to add fractions.
Type: Problem-Solving Task
Tutorials
This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting fractions. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.
Type: Tutorial
In this web-based tutorial, students learn procedures for subtracting fractions. The tutorial includes visual representations of the problems using pizzas, animations of the algorithm, and links to related lessons, worksheets, and practice problems.
Type: Tutorial
