## What should we consider first?

Fractions can be fascinating and fun and provide early access to some big ideas in algebra, so why have they been so difficult for so many students? Certainly fractions are different than whole numbers and there is added complexity to fractions. There are many different ways of thinking about fractions (part-whole, operator, measurement, number, ratio, quotient). One could say that fractions are the first big abstraction that students encounter. How then do we prepare for fractions in our early work with whole numbers and operations? Which fraction concepts and representations are most effective in laying a strong foundation as multiple meanings are encountered? This collection brings together some of the better online resources that help us think about these questions and work with students in these areas.

## How will these resources help?

There are useful summaries and discussions of common difficulties with fractions in the IES Practice Guide Developing Effective Fractions Instruction (p. 32) and in Teaching Fractions, published by the Center for Improving Learning of Fractions, and reflecting on these issues may provide insights into the early understandings that are essential. Troublesome errors include combining denominators when adding, not using a common denominator when adding or subtracting, not multiplying the denominators when multiplying fractions, confusing the whole and part in mixed numbers, and ordering fractions ignoring the denominator or treating bigger denominators as indicators of larger numbers.

While there are disagreements reflected in the selected readings about some aspects of fraction education (e.g. when to introduce formal definitions, whether the concept of ratio should be used to in developing first concepts of fraction), there is unanimity around the primacy of the unit fraction. In particular, some of the common difficulties may stem from thinking of fractions as two whole numbers, a numerator and a denominator, and this can be addressed from the beginning with a strong concept of the unit fraction as the defining element. What language, concepts, strategies, and models are used in developing a strong concept of the unit fraction?

## The "unit fraction" concept

The unit fraction concept may begin to develop with partitioning (making equal-sized parts out of a whole) and iterating (making wholes out of repeated copies of the same part) and coordinating these activities with naming practices that emphasize the unit  (fourths, thirds, etc.). Partitioning and iterating can also be used to begin to notice the relationship between the size of the part and the number of parts in a whole. This relationship depends on parts of equal size as the basis of the comparison. How can we best help students understand this comparison and the importance of "equal share"? This collection points to some virtual manipulatives that offer practice in these processes.

Some math educators and researchers such as Siebert are advocating writing out denominators as a standard practice, just as one writes out other units (3 feet, 4 seconds, 2 fifths), in order to begin fraction education with attention on the unit. A related practice is to do all sorts of counting with fraction units, emphasizing the unit, and thinking of multiples of the unit. As students notice that we make ones when the multiple is the same as the unit of the fraction (e.g. 3 thirds makes a 1), this provides early experience of the multiplicative inverse. In a similar fashion, a strong foundation in whole number place value that emphasizes counting and recording the number of different sized units can support success in developing the concept and use of the unit fraction. Counting and combining money or time are also useful early experiences with real world contexts in which students may have some ability to work with different units and to combine them through common units.

## The concept of "the whole"

The examples and models used early on must be carefully chosen to support the naming of fractional parts. If the whole is made of three cookies, how easy will it be for students to think of each cookie as a fraction rather than a whole number? How do we help students learn to name the fractional parts and use those names (e.g. 3/4 of one batch instead of 3 cookies out of 4)? When should we introduce the number line as a representation?

Hand in hand with the concept of the unit fraction goes the concept of the whole. Early success with fractions depends on consistently asking “what is the whole”? 1/4 is 1/4 OF what? It might be of a dozen eggs (3 eggs). It might be of the unit 1. It might be of 1/2 of 1 (making 1/8 of 1). In this context , one is also encountering a fraction as an operator. Coordinating the unit fraction, the whole, and the number of parts in the whole is a complex ability, but an important one that both enables students to distinguish between portion and amount, and also relates to the skills of unit analysis in word problems.

## Collection Discussion

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