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Now with the introduction of rational numbers the counting algorithm falters (that is, there is no next rational number, fractions are added differently, and so forth).
There is cause to be concerned about children's apparent lack of ability in this area.
Reporting on other NAEP findings, Post (1981) raised questions about children's ability to estimate rational numbers. In this case, the whole of which a fraction is a part, is made up of one single object such as a sheet of paper, one apple, one pie, or one rectangle. The part-whole notion of rational numbers is fundamental to the other interpretations. The part-whole interpretation is usually introduced very early in the school curriculum. It is not until fourth grade, however, that the fraction concept is treated in a substantial and systematic fashion.
Students normally explore and extend their rational number ideas through the eighth grade; then these understandings are applied in elementary algebra. Novillis-Larson (1980) presented seventh-grade children with tasks involving the location of fractions on number lines that were one or two units long and for which the number of segments in each unit segment equaled or was twice the denominator of the fraction. Results of the study indicated that among seventh-graders, associating proper fractions with points was significantly easier on number lines of length one and when the number of segments equaled the denominator.
Behr and Bright found children especially have difficulty coordinating symbolic fraction symbols with points on the number line. Such results suggest an imprecise and inflexible notion of fraction among seventh-graders. Hiebert and Tonnessen (1978) investigated whether or not the type of embodiment (continuous quantity versus discrete quantity) demands different student skills. In this study children were required to divide a quantity equally among a number of stuffed animals. We will illustrate these ideas in a later section of this chapter. Rational Numbers as Ratios The ratio interpretation of rational numbers conveys the notion of relative magnitude When two ratios are equal they are said to be in proportion to one another. This is the indicated division (or indicated quotient) interpretation of rational numbers. The major component of understanding involved in the quotient interpretation is that of partitioning. If five pizzas are shared equally among four people, how much pizza does each person get?
Draw a set of 12 circles to represent the 12 pennies, partition it to show the equal shares, and shade the circles that represent one share. What differences do you see in the partitioning that you did in Exercise 1 as compared to that in Exercise 4? 5.

In one sense its interpretation is simple; the 5 tells how many parts a whole is partitioned into while 3 tells how many of these are considered.
When the child begins to study rational numbers he or she already has a good understanding of whole numbers. The child initially understands them in terms of size, both in a relative and absolute sense. There are at least two important relationships between 3 and 5; one is additive, the other is multiplicative. The extra horizontal line represented a significant perceptual distractor for them. The children modeled these ideas using materials, pictures, symbols, and verbal descriptions. The interviews were conducted approximately every 3 weeks during the 18-week instructional period.
Analysis suggested five or six different strategies were used by children for each of the three types of conditions. The majority of these were valid strategies and in some way recognized the relative contributions of both numerator and denominator to the overall size of the fraction.
In some cases, however, children focused only on the numerator or only on the denominator and as a result made incorrect conclusions. In what ways is this a positive thing and in what ways is it a negative? Design some tasks about fractions. It seems essential that children be able to answer questions about the order or equivalence of two rational numbers to have an understanding of the meaning of operations with addition, subtraction, multiplication, and division of fractions.
Teaching, suggestions for developing this important concept are in a later section of this chapter. Another task, which proves to be a challenge to children, called Construct-the-Unit requires the child to construct the unit-whole from a given fractional part. It is the reversal of the problem of finding a fractional part of a unit-whole, and although important, it is almost never included in the elementary school fraction-related curricula. We consider this reversal task important from a point of view consistent with Piagetian psychology. Piaget suggests the understanding of a process is greater when a child is able to see that the process can be reversed to return to the starting point. In this case the process of finding a fraction of a whole can be reversed to find the whole of which a fraction is part. Another explanation suggested that the child used the given fractional part as the unit whole or used the fractional part as a unit fraction. Polhier and Sawada (1983) investigated the development of this skill from kindergarten to third grade. The basic finding of this interesting study is that partitioning ability develops gradually through a succession of five stages. A child first learns to partition in two. This is followed by the ability to perform successive halvings so that partitioning in 4, 8, 16, and so forth can be accomplished. Partitioning the whole into an odd number of parts follows when a child first observes that a cut other than that which divides in two equal parts is possible.
After this discovery is made, the ability to partition into 3, 5, 7, or an odd prime number of parts is possible for the child. For a continuous unit, one object is made into three parts, and each part is a single continuous connected entity. This singleness, continuity, and connectedness are evident to the child perceptually and therefore are likely to be conceptually evident as well. On the other hand, for the discrete set of 12 apples, each equal-sized part (equivalent subset) consists of four separate, nonconnected objects. Some children will pick out 2 thinking that that's what the numerator means, two parts. It is difficult for some children to understand initially that each part has four subparts in it.
While the research does not provide clear-cut suggestions about this matter, we recommend the fraction could be developed on the basis of continuous models and then a transfer can be made to discrete models. We also recommend children be given the opportunity to partition various objects.
For example, partition a sheet of paper into two equal parts, later into four equal parts, then eight.
Young children will need some special guidance in partitioning paper into an odd number of parts. Where to make the first cut, other than in the middle, needs to be given special attention.
Similarly, children can be given sets of 4, 8, or 12 counting chips and asked to partition these sets into four parts.
After a child has made the partitioning into four parts, an important activity is for the child to designate (that is, show the teacher and other children) what is one part, two parts, three parts, and four parts. Notice that one doesn't have to bring in the language of fraction for a child to have a meaningful experience with partitioning. These early experiences with partitioning may be as important to a child's development of fraction concepts as counting is to their development of whole number concepts.
Ask the children to fold the sheet in either of the two ways suggested by a dotted line as shown in Fig.
After the second fold is made in accordance with one of the patterns shown in Fig.
Then the child should be asked to show one, two, three, and four parts. A challenge activity would be to continue folding any one of the results shown in Fig. Thus, even as the basic concept of fraction is being developed, the concept of adding fractions with like denominators can be foreshadowed. Chips can be used as a discrete model.
7-l8). Not all children will necessarily know that to partition a set of six chips into three equal subsets two chips go into each group. To help you might suggest that the child start three sets by moving three chips, each to a separate spot (see Fig. Then a child can put more chips into each subset, one at a time, until all chips are used up (Fig.
Finally, two parts (two subsets each with two chips) are covered with different colored chips (Fig. It should be a rod that can be partitioned by another rod (evenly) into four parts. 7-22). Having chosen a unit, a child may have to experiment to find a rod to accomplish the partition shown in Fig. 7-23. Some children may experiment with a three-rod or a one-rod before discovering a two-rod will work. We need to fold another region in order to show 5 one-fourth parts. Step 3.
It has to do with an important relationship between the size of each part and the number of parts when two unit-wholes of the same size are partitioned into 8 and 5 parts respectively. For some fractions in this category children invent (and can be taught) interesting strategies.
And also observe how important the ability to order (determine which was larger) unit fractions appeared to be. The range of its applicability is subject to having well-understood reference points.
Children must develop a meaningful algorithm or procedure for determining which fractions are equivalent fractions. With children we usually limit n to whole numbers although there are long-term advantages to allowing n to be a fraction or mixed number as well.
Is it possible that the difficulty that is caused is also an appropriate situation to facilitate learning? This is highly desirable. Paper folding is useful for developing concepts that underlie addition and subtraction of fractions with unlike denominators. A long narrow slip of paper (for example, adding machine tape) works well for this activity.
The relative error due to folding inaccuracies will be small and frequently not noticeable.
7-39. To tell the sum, we need to know what fraction of the whole strip is shaded. The read should have centimeter rods at hand and follow the steps while reading. Step 1.
Build rod trains of two-rods and five-rods side by side until both trains are the same length (Fig.
The Construct a Sum task was designed to elicit insights into the development of qualitative (nonalgorithmic) approaches to a variety of fraction understandings in fourth- and fifth-grade students. Children were asked to select digits to form two rational numbers whose sum was as close to 1 as possible. High achievers were flexible and spontaneous in their conceptions of fraction order and equivalence. Low performers tended to apply concepts in a constrained and inflexible manner. We encourage you to try this activity with a classmate first and then with several intermediate-level children. Some writers and teachers argue that multiplication should be taught to children before addition and subtraction because of this fact. While the algorithm for multiplication is simpler, the question of whether multiplication is conceptually simpler is not so easily resolved. It is well known from research that children are able to perform additive operations before they can perform multiplicative operations. For whole numbers this is probably true because one of several interpretations for multiplication is that of repeated addition. From a manipulative perspective, it appears the operation of multiplication has the more complex manipulative base.
For this reason we suggest multiplication be introduced after addition and subtraction. To do this, we complete the partition of the unit-whole (see dotted lines in Fig. 7-48). Now observe that the region marked with is 8 of 15 parts. 7-48 and observe that the eight parts appear as two rows of four; that is, 2 x 4. Children can solve this problem by distributing (like dealing out cards to players) the 12 cookies 1 by 1, 3 at a time, or 2 cookies to each child in one round and 1 in another to give each child an equal share of cookies. Another way to think of this is: How many groups of 4 cookies are there in a group of 12 cookies? This type of division is sometimes related to repeated subtraction in terms of the number of subtractions of 4 until a remainder of zero is obtained: 12 - 4 = 8, 8 - 4 = 4, 4 - 4 = 0. It took 3 subtractions to get to zero, or 3 groups of 4 cookies were removed to use up all of the cookies, so 3 children can be given 4 cookies. For partitive division of rational-numbers, we will find it convenient to change one or both of the fractions so they have a common denominator before we carry out the equal distribution of dividend unit fractions among the unit fractions of the divisor.

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