338 MatheMatics teaching in the Middle school ● Vol. 15, No. 6, February 2010Kathleen Cramer,Debra Monson, Stephanie Whitney, Seth Leavitt, and Terry WybergDividing Fractions andiSToCKphoTo.CoMCopyright © 2010 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.iInvert and multiply. Fraction division is generally introduced in sixth or seventh grade with this rule. We examined current com-mercial curricula and found that few textbooks use context as a way to build meaning for the division of fractions. When context is used, the connection between the invert-and-multiply rule and the context is superficial at best. Textbooks often use illustrations as a form of representation to build meaning. However, the transition from these pictures to the symbolic rule occurs quickly. In so doing, students may be getting an inadequate understanding of fraction division. One way that textbooks show division with fractions is by pro-viding a picture for a single problem, such as 3 ÷ 1/2. In this case, students would be shown three rectangles, each divided in half, so that the solution of 6 is recognized (see fig. 1a). The invert-and-multiply algorithm is presented after this one example. It is likely that the pictures will appear once or twice in the practice set fol-lowing the example.See how a class of sixth graders used concrete and pictorial models to build meaning for arithmetic operations with fractions.Vol. 15, No. 6, February 2010 ● MatheMatics teaching in the Middle school 339Dividing Fractions andproblemSolving340 MatheMatics teaching in the Middle school ● Vol. 15, No. 6, February 2010Another common textbook meth-od is exemplified by the example of 89291892948929899289241118992÷÷=÷=×÷//×//..figc99A rectangle is divided into nine pieces, eight of which are shaded to represent 8/9 (see fig. 1b). The caption, or explanation, in the book would group the sections two at a time, into four portions, yielding the result. The symbolic representation and algorithm of the example follows (in fig. 1c). Each of these examples demonstrates a quick transition to a symbolic rule.Our approach differs from the way that fraction division is introduced in traditional curricula. We presented fraction division as a problem-solving opportunity for students. Students constructed their own strategies that relied on pictures. Their strategies and pictures reflected their under-standing of the part-whole model for fractions and their ability to name fractional parts when the unit changed. The story problems we used were based on a measurement model for division. Most textbooks tend to use a measurement model for division (van de Walle 2007). Similar contexts can be used to write both measurement and partitive models. Consider the follow-ing two examples:������������(a)Three wholes divided into halves is a way to represent 3 ÷ 1/2.������������(b)8929489298992892941118992÷=÷=×÷//×// is shown by shading 8/9 and combining into four groups.The problem: 8929489298992892941118992÷=÷=×÷//×//Multiply by the reciprocal: 8929489298992892941118992÷=÷=×÷//×//Multiply and simplify: 41118992414//×//==(c)Fig. 1 Fraction division is introduced in commercial middle school textbooks in a variety of ways.STeVe SNyDer/iSToCKphoTo.CoM 1. Kay has an 8-gallon bag of soil for her garden plants. If each planter uses 2/3 gallon, how many planters can she fill? 2. Kay has an 8-gallon bag of soil for her 12 planters. How much soil will be in each planter?The first problem is an example of a measurement model; students are given a set measure, 2/3 gallon, and they must determine how many groups of 2/3 gallon are in the 8-gallon bag of soil. The second problem is an example of a partitive model, because students know the number of groups and are determining the amount in each group.We chose to use a measurement model because we anticipated that it would give sixth-grade students an opportunity to solve fraction divi-sion problems using pictures. In so doing, this choice would support the common-denominator procedure for fraction division. Since these students had already learned to use common denominators to add and subtract fractions, the measurement model for fraction division gave them another opportunity to see how common de-nominators can be used when operat-ing with fractions. We acknowledge that this four-day problem-solving experience represents students’ initial entry into division with fractions. Future instruction should include the parti-tive model so that it can support their construction of the invert-and- multiply procedure.The numbers in the story prob-lems varied; they included a whole number divided by a fraction, division of two fractions, and answers with and without remainders. The strate-gies that students constructed builtVol. 15, No. 6, February 2010 ● MatheMatics teaching in the Middle school 341meaning for dividing fractions using a common-denominator approach.students’ BackgroundTo better understand why students were able to construct their own strategies for solving division prob-lems using fractions, we share some background on what happened before students worked with division. These sixth graders first completed a four-lesson review of fractions. They used fraction circles and paper folding to reinforce their understanding of part to whole. The lessons emphasized the importance of identifying the unit when naming fractions. An early activity used the repre-sentations shown in figure 2. Stu-dents named fraction-circle pieces in many ways: the blue piece was 1/4 when the black circle was the unit and 1/2 when the yellow piece was the unit. The red piece was 1/12 when the black circle was the unit, 1/6 when the yellow piece was the unit, and 1/3 when the blue one-fourth piece was the unit. The pur-pose of this activity is to reinforce the part-whole construct for fractions and help students understand the impor-tant role that the unit plays in naming a fractional amount. Students see that each piece of the circle can be named in more than one way depending on what is stated as the unit. Understanding the role of the unit is critical when using pictures to divide fractions. Readers will see how this idea plays out later in fraction di-vision when examining