conceptualizing division with remainders

426 March2012•teaching children mathematics www.nctm.org

Copyright © 2012 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.

By Teruni Lamberg and Lynda R. Wiest

Third graders’ struggles to solve contextualized, student-generated division problems with

remainders offer insights to guide instruction.

www.nctm.org teaching children mathematics • March 2012 427

Conceptualizing

What do you do with the remainder when you divide?” Mrs. Thompson asked her third-grade students. They replied with such comments as, “You can’t share

that, because they won’t be equal!” and “It’s not going to come out even because you can’t do that!” These answers were consistent with third- and fourth-grade student performance in a pretest and a posttest admin-istered as part of a math professional development project conducted by author Teruni Lamberg. In this test, most students successfully solved the division problem involving whole numbers but were unable to solve 27 ÷ 4, which made the authors wonder why chil-dren have difficulty with remainders.

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What is involved in helping children under-stand simple division and division with remain-ders? Reys and colleagues found that “children should encounter division problems involving remainders from the time they begin to work with division ideas…. As long as problems remain on a concrete level, the concept of remainder is rather easy” (2007, p. 277).

This article explores a lesson Thompson conducted with her third graders that involved using concrete materials to reason informally about division with remainders. The lesson was part of a unit on division that Thompson had developed.

The division-with-remainders taskThe purpose of this lesson was to have students explore the meaning of division using their own informal problem-solving methods. Thompson first activated prior knowledge about division by discussing scenarios that do and do not require division. Students then worked in designated groups with three or four classmates to write and solve a division word problem using objects supplied to them as inspiration for the problem

context. NCTM (2000) promotes such contextu-alized learning. Considering remainders in con-text and handling them accordingly is especially important (Reys et al. 2007; Van de Walle, Karp, and Bay-Williams 2010).

Thompson assembled four different bags of objects, one for each student group:

• Bag1: 100-centimeter tape measure • Bag2: 55 pencils• Bag3: $56.88 in play money• Bag4: 75 square tiles

The objects in the bag could represent anything, but problem contexts were constrained by the given materials. Having to create and solve a division problem required students to think about the meaning of division.

Students had to begin by deciding which divi-dend they wanted to use. Open-ended problems such as this require reasoning. Although divi-sion may be conceptualized in more than one way structurally, children tend toward the fair-sharing (partition) model, perhaps because they find the concept of separating a given number of items into “fair shares” more comprehensible or because they are given more experience with this context. (The other major type, measure-ment or quotitive division, involves distributing a given amount of something to an unknown number of groups, which is what must be deter-mined. For example, if there are twenty-four dice and each child needs four to play a certain game, how many children can play?)

Student efforts to solve the division taskStudents worked together in their small groups to write a division word problem that “fit” the bag of materials they were given. Almost all groups started to solve their problem by attempting to physically distribute the material in each bag equally among their group mem-bers. In other words, the number of children in the group automatically became the divisor. The children uniformly chose a fair-shares model in which the provided material was an amount (the dividend) to be shared equally among group members.

Most groups started physically distributing the items from the bag and then determining the total quantity (dividend) they had. When

third-grade students symbolically represented their problem situation correctly using a number sentence. they made tally marks in two groups to check their answer.

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students discovered that the concrete objects could not be divided equally, most of them piled the objects in the middle of the table and started redistributing them. Nearly all the children had to manipulate the concrete objects to solve the problem. They seemed to understand that one key conception of a division situation involves separating objects into equal groups. This physi-cal action was critical for students to actively consider the meaning of division and create a visible record of their efforts.

Group 1: Tape Measure problemStudents in group 1 had a 100-cm measuring tape. They initially decided to share the tape equally between two students because this was an easier problem to solve. They folded the tape in half and said that each of the pair would get 50 cm. This division action was physically easy to do. Folding allowed the children to see that the pieces were congruent, or equal in length. Thompson challenged the group to think about items that can be measured in length, such as ribbon. Group 1 created a scenario of two chil-dren who were to share yarn that could be cut into smaller 10-cm pieces to make bows for boxes. Now the children had more difficulty partitioning the unit. James argued that the distance between 40 and 50 cm is only 9 spaces long because the ends should not be counted. Eventually the group decided to place markers at intervals of 10, requiring students to think about how to use a number line to identify inter-vals of 10. For example, students had to count the interval from 40–41 as 1 cm. Placing markers in intervals of 10 allowed students to keep track of their counting. They created a problem that involved more sophisticated thinking; solving the problem required two steps:

There is a 100-centimeter piece of yarn. Two children wanted to make bows, so they split the yarn into 10 centimeter pieces. How many pieces do they each get?

The 100-cm piece of yarn represents the whole. Students had to think about how to equally split the 100-cm piece of yarn between two students who were each to get the same number of 10-cm pieces.

Students solved the problem and symboli-cally represented the problem situation cor-

Differentiating Division Lessonsthis lesson can be adapted to different levels of sophistication in the elementary school classroom. for example, a teacher can adapt the provided problem contexts to match class ability levels, interests, and needs. teachers must support students’ written records and build on current understanding of division to move students toward more efficient strategies and more complex thinking. implementing this lesson offers the following explorations.

• Whatdoes division mean? What do the divisor and dividend represent?

• Whyand how do you make equal-size groups when you divide?

• Whatefficient strategies can we use to record and solve division problems?

• Whatdo you do with a remainder? When is it appropriate to divide the remainder into fractional parts, and when does it make more sense to round up or down to the nearest whole number?

• Howcan you check answers to division problems?

a group created a division problem using 100-cm tape. they placed markers at intervals of 10 to keep track of their counting.

rectly using a number sentence (see fig. 1). The tally marks and the equation 100 ÷ 2 = 50 cm represent how they originally thought about the problem context. They made tally marks in 2 groups to check their answer. They recorded the intervals of 10 they had identified with their markers by using tally marks to organize the groups of 10 for easy counting. They also recorded 100 ÷ 10 = 10 and wrote that each group will get 5 (10-cm) pieces. Furthermore, they represented the answer by showing how tally marks in groups of 5 can be divided into 2 groups of 50.

Group 2: Pencil problemThe students in the second group emptied the pencils from their bag into the middle of a table. A student distributed the pencils 1 at a time to each of the 4 group members in an attempt to divide the whole unit of 55 pencils evenly.


Students kept getting frustrated because they could not make this happen. When they com-pared the quantities of the pencil groups and found them to be uneven, they kept return-ing pencils to the middle of the table and re-counting them without determining the total number of pencils they had. Finally students realized that 3 group members had 14 pencils and 1 member had 13. The 3 students returned 1 pencil each, but they eventually took them back and told the fourth student that he should be content with 13. In the long run, the group was so uncomfortable with the uneven groups that they eliminated the remainders by hiding the leftover pencils. The problem they created was thus a “sanitized” version of the original (see fig. 2). The students’ recordings show a transition from a repeated-addition approach (4 groups of 10 + 3) to a multiplicative strategy that involved calculating 4 × 10 = 40 and 4 × 3 =12 and then adding the results.

Group3:Dividing75squaretilesThe third group decided that their tiles repre-sented books and thus used the tiles symboli-cally. Separating the tiles into 4 equal groups of 18, students discovered they had 3 leftover. Tommy decided they could divide these tiles further, giving everyone a one-half tile, and the remaining tiles could be further divided into quarters. Although the remainder can be subdivided mathematically, this does not make sense in the context of books. When Thompson asked the group if they could really divide the books, they changed their minds and stated that the tiles no longer represented books. They changed the problem context to apples, which could be further divided. In the end, the group eliminated remainders completely and rewrote the problem as 75 ÷ 5 (see fig. 3). Their recordings show that the students used succes-sive approximation to solve the problem. They made 5 circles to represent children, and they distributed the apples to the children in groups of 5 apples each until they reached their total number of apples (75). They added the num-ber each child got and found it to be 15 apples apiece. Students checked their answer by add-ing 15 five times. They then symbolically repre-sented the problem using a number sentence (75 ÷ 5 = 15). Furthermore, they had considered the meaning of a remainder in their particu-

after students in group 3 created a division problem with square tiles to represent books, they realized that apples would work better than books for fractional parts.

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Group 2 found remainders so awkward to deal with that they hid their extra pencils to create a cleaner version of the original problem.

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lar problem context (fair-shared books) once prompted by the teacher to do so. Students in this group were somewhat uncomfortable with a fair-sharing problem involving a remainder that could not be partitioned, so they changed the problem to one that could be divided evenly. This problem provides a good oppor-tunity to discuss as a class what to do with a remainder: When does it make sense to split a remainder into fractional parts, and when does it not?

Group4:DividingmoneyThis group was given the task of writing a divi-sion problem for $56.88 worth of play money. The denominations included two $20 bills, one $10 bill, one $5 bill, one $1 bill, 3 quarters, 2 nickels, and 3 pennies. The group wanted to distribute the money evenly among its 4 members. However, students found that dividing a $20 bill and a $10 bill among 4 chil-dren is physically impossible using the given denominations. They kept trying to give each child the same amount of play money but were constrained by the denominations. Thompson

the children in group 4 did not automatically assume that everyone would get the same amount if they distributed money evenly.

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told them that they could exchange the bills for comparable values using different play money denominations. At this point, students were able to successfully distribute the money among 4 students, and each counted his or her portion. Despite some initial “false starts,” the group was ultimately able to symbolically record its problem correctly (see fig. 4).

Follow-up instructionHer students’ informal explorations and prob-lem development during these tasks gave Thompson much insight into their thinking and how to proceed with subsequent instruction. Students clearly needed much more experience with division problems with remainders so they could become more comfortable and skilled with solving them. Such skill includes the ability to consider real-world contexts in determining answers to problems. Accordingly, Thompson devoted follow-up instruction to solving prob-lems she posed and having students do similar tasks to those described in this article until stu-dents achieved a reasonable level of confidence and mastery. She posed problems that required the exact remainder to be retained (as in How many ounces of soda do each of 4 children

early exposure to the concept of division with remainders helps students circumvent confusion.

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get from 34 ounces?), problems in which the remainder should be discarded (as in How many 7-inch bracelets can be made from a 38-inch chain?), and problems in which the remainder requires the answer to be rounded up to the next whole number (as in How many buses that hold 60 students each are needed to transport 215 third graders on a field trip?). The materials to be divided included different types of units based on discrete or continuous models (e.g., cook-ies versus pie). Over time, student-constructed problems started to show similar variety.

Classroom implicationsTeaching division effectively involves building a bridge between children’s intuitive models of division and the formal operations that rep-resent children’s conceptual understanding. Analyzing students’ reasoning is important teacher feedback for helping children make meaning of situations and progress in their mathematical thinking. This is why the les-son was designed to have students first think explicitly about division situations using intuitive models. Intuitive thinking is central to number sense in the early years, which is foundational to fluent whole-number compu-

tation (Fennell 2008). The goal was to have this informal understanding lay the foundation for formal mathematics. Anghileri, Beishuizen, and Van Putten (2002) found that contextual-ized problems are more conducive to informal strategies founded on number sense, whereas bare number problems are more likely to be attempted using an algorithm. The research-ers found that although informal methods may be inefficient, supporting those meth-ods initially and helping scaffold students to more efficient mathematical procedures led to greater cognitive gains and fewer miscon-ceptions than using formal methods alone. In the lesson described here, having students generate problems by manipulating objects provided a more concrete entry for students to build on their intuitive knowledge, engage in discussion, and think in more complex ways.

The open-ended task of generating division word problems forced the children to think about the meaning of division. Students devel-oped and solved a problem and put their prob-lems and solutions into writing. They needed to understand the meaning of the dividend and divisor. Further, they explored the physical act of partitioning objects to make equal-size

having meaningful work generating contextual word problems leads students to better understanding of division with remainders and less reliance on algorithms.


groups. During this process, some were faced with remainders. Some groups struggled with this situation and changed the problem details to create a problem that resulted in no leftovers. This shows students’ need for much meaningful work with division problems with remainders to develop facility and comfort with making reasonable real-world decisions about how to handle them in varied contexts.

REFERENCESanghileri, Julia, meindert Beishuizen, and kees

Van putten. 2002. “from informal Strategies to Structured procedures: mind the Gap.” Educational Studies in Mathematics 49 (february): 149–70.

fennell, francis. 2008. “number Sense—right now!” NCTM News Bulletin 44 (march): 3.

national council of teachers of mathematics (nctm). 2000. Principles and Standards for School Mathematics. reston, Va: nctm.

reys, robert, mary m. lindquist, Diana V. lambdin, and nancy l. Smith. 2007. Helping Children Learn Mathematics. 8th ed. hoboken, nJ: Wiley and Sons.

Van de Walle, John a., karen S. karp, and Jennifer m. Bay-Williams. 2010. Elementary and Middle School Mathematics: Teaching Developmen-tally. 7th ed. Boston: allyn and Bacon.

Teruni Lamberg,[email protected], is an associate professor of elementary mathematics education at the Univer-sity of nevada in reno. her research interests include children’s mathemati-cal thinking, teacher education, and technology in mathematics education. LyndaR.Wiest, [email protected], is a professor of mathematics education at the University of nevada in reno. her

scholarly interests include mathematics education, educational equity, and teacher education.

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