## 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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Created:04-10-2012 by steve
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Created:06-27-2012 by Uncle Bob
Last Post:07-16-2012 by bethb

## Resource Title/Description

This practice guide presents five recommendations intended to help educators improve students’ understanding of fractions. Recommendations include strategies to develop young children’s understanding of early fraction concepts and ideas for helping older children understand the meaning of fractions and the computations involved. The guide also highlights ways to build on students’ existing strategies to solve problems involving ratios, rates, and proportions.
This research guide provides suggestions for specialists and teachers looking to improve fraction instruction in their classrooms or schools. The guide starts with ideas for introducing fraction concepts in kindergarten and early elementary school and continues with activities and teaching strategies designed to help older students understand fraction magnitudes and computational procedures involving fractions. It then examines ways of helping students use fractions to solve rate, ratio, and proportion problems. Each recommendation includes a brief summary of supporting research and descriptions of classroom activities that can be used to implement the recommendation.
Teachers can use this interactive tool to help students build a conceptual understanding of fractions by linking visual models to numeric representations. Students learn about equal parts of a whole (denominators) and shaded parts (numerators). This page includes a video demonstration of the tool and sample lessons from the Conceptua curriculum. Free registration is required to use the tool. A paid subscription is necessary to access full curriculum and allow full student use.
This interactive Flash applet develops children's fraction concepts (partitioning, equivalence, and unit) by building on their informal understanding of rational numbers in the familiar context of sharing cookies fairly among friends. Users decide on the number of friends, the shape and number of the cookies, and how to cut and distribute them equally. The activity is very flexible and incorporates multiple representations of fractions, area models, numerical symbols, and voice cues. Downloadable pdfs of cookie images are provided as well as a User Manual that includes implementation suggests and a review of research.
This exploratory problem provides students a way to consolidate their understanding of halving and halves and gives students experience of mathematical proof. The students are given multiple images of squares split in half. The goal is to prove how they are correctly halved and to think of other ways to split a square into two halves. The Teachers' Notes page offers rationale, suggestions for implementation along with a PowerPoint presentation, discussion questions, ideas for extension and support, and printable (pdf) worksheets of the problem.
This four-page article describes four basic approaches children use in understanding fractions as equal parts of a whole. Topics covered include working with equal shares, partitioning regions and units, understanding of fraction equivalence, and ideas for teaching fractions more effectively. References are given.
Teachers can use this interactive tool to help students develop fraction sense by arranging pattern blocks to explore unitizing. Then they can see their correct answer tessellate. This page includes a video demonstration of the tool. Free registration is required to use the tool. A paid subscription is necessary to access full curriculum and allow full student use.
This 5-minute video captures a lesson in which students create pattern block designs and use them to explore part-whole relationships. The teacher explains the purpose and implementation of the lesson, the role of manipulatives, and how the activity invites multiple representations. Included are reflection questions for the viewer and a downloadable transcript (doc). Registered users (free) may download the video.
In this lesson plan from Illuminations, students use relationship rods to explore fraction relationships. Relationship rods range in length from one to ten centimeters, and each rod is a different color. An activity sheet with solutions, questions for students, assessment options, and suggested extension activities are included. The lesson plan is part of a five lesson plan unit, Fun with Fractions, which is cataloged separately.
This problem helps develop an understanding of the relationship between the part and the whole. Given a square figure divided into smaller triangles, students are asked to use the pattern to divide the square into two halves, three thirds, six sixths, and nine ninths. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, ideas for extension and support, and a printable (pdf) worksheet of of the problem.
In this lesson, students will explore fractional relationships using pattern blocks and explain how relationships change when the one (unit) changes. The lesson includes "Connecting Learning" questions and worksheets to make a "What is the One?" book.
This blog page offers more than 25 activities and games that use a Hundred Chart to develop students' skills and concepts in a variety of math topics: basic operations, number sense, patterns, number theory, fractions, decimals, and logic. The suggestions include links to materials and to other websites.