Operation E

Operations with Fractions: Addition and Subtraction

Fraction Learning Pathway

Add and subtract fractions with unlike denominators (e.g., 2 and 7) using models and symbols

Using models to visually and physically compose and decompose fractions with common units supports students in understanding the roles of the common unit (or common denominator) in addition and subtraction.

Background

Prior to more formal exposure to fraction addition and subtraction, students need a solid understanding of fractions as quantity, as well as part-whole constructs of fractions (Petit, Laird & Marsden, 2010). A strong foundation in equivalence is also crucial to student understanding of addition and subtraction with fractions (Petit, Laird & Marsden, 2010) . When fluency with equivalent fractions is developed, students are better able to consider addition of unlike fractional units by first relating each quantity to a common unit (common denominator) (Empson & Levi, 2011). When students develop an understanding that the need for a common unit is universal for all addition and subtraction, they can more readily connect their understanding of whole number addition to other number systems, such as decimals and fractions, as well as algebraic operations. This increases student fluency of addition and subtraction across all number systems.

Junior grade students should be exposed to tasks that allow them to understand fraction operations in connection to whole number operations, beginning with the provision of ample time to allow students to construct their own algorithms for the operations (Huinker, 1998; Brown & Quinn, 2006). By focusing on sense-making early on, rather than memorization of an algorithm, students will be able to extend this learning into algebraic contexts in secondary and post-secondary studies (Brown & Quinn, 2006; see also Wu, 2001) and build fluency with meaning.

Several studies have shown “that if children are given the time to develop their own reasoning for at least three years without being taught standard algorithms for operations with fractions and ratios, then a dramatic increase in their reasoning abilities occurred, including their proportional thinking” (Brown & Quinn, 2006, p. 5, citing Lamon, 1999).

Specific connections to addition and subtraction with whole number should be made as the properties hold true for operations with fractions as well.

  • The Commutative Property: a + b = b + a
  • The Associative property: (a + b) + c = a + (b + c)
  • The Identity Property: a + 0 = a; a - 0 = a
  • The Distributive property: a(b + c) = ab + ac; a(b - c) = ab - ac
As well, students should understand that addition in the inverse of subtraction and vice versa. That is to say that if a + b = c, then a = c - b and b = c - a.

Students benefit from intentional use of models when learning about the operation with fractions. The following two examples help us to see how student strategies for adding fractions can build from intuitions and familiarity with whole number operations. In these examples, students use models to add 25 + 15 .

A composition strategy:

A fraction strip with one fifths shaded in and another fraction strip with three fifths shaded in.

Students may recognize that both fractions are referring to fifths and may choose to create a whole unit and partition it into fifths. They could then shade in each of the fractional values to determine what fraction of the whole is shaded. In this way, the student can see that 25 + 15 = 35.

An additive strategy:

A fraction strip with two fifths filled in with full circles and one fifth filled with empty circles

Students may identify 1 one-fifth unit and then iterate it to create the two fractions. They would then consider what fraction of the whole is shaded, possibly by holding a visual of the unrepresented fraction of the whole. In this way, the student can see that 25 + 15 = 35.

References:

Brown, G., & Quinn, R. J. (2006). Algebra students’ difficulty with fractions: An error analysis. Australian Mathematics Teacher, 62, 28-40.

Empson, S. & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals: Innovations in cognitively guided instruction. Portsmouth, NH: Heinemann. Pp. 178-216.

Huinker, D. (1998). Letting fraction algorithms emerge through problem solving. In L. J. Morrow and M. J. Kenny (Eds.), The Teaching and Learning of Algorithms in School Mathematics (pp. 198-203). Reston, VA: National Council of Teachers of Mathematics.Lamon, S. (1999). Teaching Fractions and Ratios for Understanding. Mahwah, N.J.: Lawrence Erlbaum Associates.

Moss, J. & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education 30(2), 122-147.

Petit, M., Laird, R., & Marsden, E. (2010). A focus on fractions. New York, NY: Routledge.

Tasks

Creating Sums Close to One

Students will explore sums of fractions to determine two fractions with unlike denominators that, when added, will be close to but not equal to one. This task encourages the student to develop the sense of fractions as quantities (fundamental to fractions understanding) and also to justify their thinking by representing their mathematics in a variety of ways.

Creating Sums Close to One Task

Building to Operations with Relational Rods

This bundle uses the foundational concept of unit fractions to help students compare fractions and build understanding of adding and subtracting fractions with friendly or unlike denominators using models and symbols. Students with fragile understanding of unit fractions often have difficulty transitioning to adding and subtracting fractions. The tasks in this lesson bundle encourage students to be flexible in recognizing that a whole can be any length.

Equals Game

This is a series of games to be played in pairs, which will take students from building and naming equivalent fractions, through to building and naming addition and subtraction equations. Students compose and decompose fractions in order to add and subtract using cards (fractions, equals sign and operation cards) and relational rods.

Equals Game Task

Making Mixed Numbers

This task was designed to move students from representing a mixed fraction towards using operations with fractions, and is a suitable exploration for students even prior to the formal introduction of these concepts. The task consists of a series of prompts intended to scaffold students to use multiple representations to visualize and conceptualize the problems.

Making Mixed Numbers Task

Turf Touchdown

Students represent the installation of turf, which is installed at different fractional rates each day, in a newly constructed field. By creating a model and determining common fractional units, they will be able to establish how much turf needs to be installed on the final two days.

Pouring Container showing one-fourth

Composing and Decomposing 178

In this hands-on task, each student independently represents 178 by creating either a physical model (e.g., using tiles, folded paper, relational rods, or other items with which students are comfortable working) or a two-dimensional model (e.g., a drawing or a number line) of their choice. Students are then challenged to create as many addition and subtraction number sentences as possible to represent 178

Pouring Container showing one-fourth

Additional Prompts

These tasks emerged out of the fraction research. Teachers may wish to use them as diagnostic or summative assessments, exit cards, number talk prompts, or additional practice questions. By considering both the specifics of the cell and student use of purposeful models, teachers can support students in acquiring a strong conceptual understanding.

Curriculum Connections