Progressing to friendly but unlike denominators allows students to further partition one of the fractional units to generate the other unit in order to combine or subtract the quantities.
BACKGROUND
Prior to more formal exposure to fraction addition and subtraction, students need a solid understanding of fractions as quantity, as well as partwhole 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 fluencyof 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 sensemaking early on, rather than memorization of an algorithm, students will be able to extend this learning into algebraic contexts in secondary and postsecondary 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). 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
2
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+
1
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 A composition strategy: Recognizing that both fractions are referring to fifths, a student 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
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5
+
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/
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=
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 An additive strategy: A student may identify 1 onefifth 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
2
/
5
+
1
/
5
=
3
/
5
.
References:
Brown, G., & Quinn, R. J. (2006). Algebra students’ difficulty with fractions: An error analysis. Australian Mathematics Teacher, 62, 2840.
Empson, S. & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals: Innovations in cognitively guided instruction. Portsmouth, NH: Heinemann. Pp. 178216.
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. 198203). 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), 122147.
Petit, M., Laird, R., & Marsden, E. (2010). A focus on fractions. New York, NY: Routledge.

TASKS
Building a Stage
Students will determine an unknown measure by using a tape measure as a number line. Within the context of building a stage using layers of different types of wood, students will find the sum of friendly but unlike denominators. This will help them to understand the need for common units (denominators) when adding fractions.
Relay Race
Students are introduced to the concept of a relay race where participants complete various fractional distances of the total. Based on two known fractional distances of a race with unlike denominators, students will determine the distances that the other runners might complete.
The Flick Game
Students will explore equivalent fractions and sums of fractions as they flick an eraser along game boards that are divided into thirds, sixths, ninths and eighteenths. The process of recording the scores on a number line – and determining the sum of their personal score, as well as their team score – reinforces the need for a common unit for addition of fractions.
