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Number & Operations for Teachers Copyright David & Cynthia Thomas, 2009 |
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Modeling the Meaning of
Addition and Subtraction Fraction
Bar Model Fraction bars
are commonly used as concept models for part/whole relationships. For instance, Figure 4.1 shows three
fraction bars representing the fractions 1/2, 1/3, and 2/5. By comparing the shaded portions of these
fractions, it is clear that 1/3 < 2/5 < 1/2.
Figure 4.1: Fraction Bars Modeled using
fraction bars, equivalent fractions
are recognized by the fact that the shaded portions of their respective
fraction bars are equivalent in length.
Figure 4.2 presents a concept model and an algorithm for generating
fractions equivalent to 1/2 and 1/3.
Clearly, this process could be extended indefinitely, generating as
many equivalent fractions as desired.
1/2 =
Figure 4.2: Equivalent
Representations for 1/2 and 1/3 Using
the concept of equivalent fractions, we may explain both the concept and the
algorithm for fraction addition. For
instance, in Figure 4.3 fraction bars representing 1/2 and 1/3 are shown
beneath a fraction bar of unit length.
A bar divided into sixths is also shown. Note that the shaded portion of the
fraction bar representing 1/2 is the same length as 3 of the cells in the
last row. The interpretation of this
finding is that 1/2 is equivalent to 3/6.
Comparing the bottom two fraction bars leads to the finding that 1/3
is equivalent to 2/6. Since the
fractions 3/6 and 2/6 have the same denominator, the number 6 is often called
a common denominator. Another way to express this thought is to
say that 1/2 and 1/3 have a common
divisor, 1/6. This statement
emphasizes the fact that 1/6 divides both 1/2 (three times) and 1/3 (two
times).
Figure 4.3: Common Denominator/Divisor Because
both 1/2 and 1/3 may be expressed in terms of sixths, the sum 1/2 + 1/3 many
be re-expressed as 3/6 + 2/6. This sum
in represented graphically in the third row of Figure 4.4.
Figure 4.4: Modeling Fraction
Addition using Fraction Bars Figure
4.5 shows three additional solutions based on the use of alternative common
divisors. In the first case, a common
divisor of 1/12 is used. In the second
case, 1/18 is used as a common divisor. And in the third case, the common
divisor is 1/24.
Figure 4.5: Equivalent
Solutions based on Alternative Common Divisors Figure
4.6 presents the same solutions in the context of a systematic search for
common denominators. Notice that every
common denominator corresponds to a fraction that is a common divisor of both
1/2 and 1/3.
Figure 4.6: Finding Equivalent
Common Denominators Figures 4.2 and 4.6 may be used to explain why the
standard “rule” for finding a common denominator always works, that is, why
you multiply each fraction, numerator and denominator, by the denominator of
the other fraction. Beginning at the
top of the column “Fractions Equivalent to 1/2” in Figure 4.2 and reading
down, it is clear that every integer ratio n/n (n ≠ 0) will
eventually be used as a multiplier of the fraction 1/2. In particular, the denominator of the
second fraction, 1/3, is used to form the fraction Example 4.1 Using
fraction bars, model the sum Solution 4.1
Area
Model Another useful model for explaining fraction addition is the area model. In Figure 4.7, a unit square is partitioned two different ways. In the partition on the left, the shaded portion represents 1/2 the area of the unit square. In the partition on the right, the shaded portion represents 1/3 the area of the unit square.
Figure 4.7: An Area Model for Fractions In Figure 4.8, the two partitions represented in Figure 4.7 are superimposed on one another. This superposition results in a new partition of the unit square. [Note: The cell in the upper-left hand corner is shaded darker than the rest because two shaded cells overlap there, each representing 1/6 of the area of the unit square.] The new partition suggests a relationship between the two original partitions. The horizontal lines divide into thirds the shaded region representing 1/2 the area of the unit square. And the vertical line divides in half the shaded region representing 1/3 the area of the unit square. Each of the new areas created in this manner has the same area, 1/6 that of the unit square. Furthermore the region with area 1/2 has the same area as three of the smaller regions. In other words, 1/6 divides 1/2. Using similar reasoning, we may conclude that 1/6 also divides 1/3. So, 1/6 may be seen as a common divisor of the fractions 1/2 and 1/3 and 6 may be seen as a common denominator.
Figure 4.8: Superposition of Two Partitions Figure 4.9 summarizes these findings, repositioning one of the overlapping areas superimposed in the upper-left hand corner of the figure. Doing so facilitates the counting of cells representing the common divisor, in this case, cells with area 1/6. This process is also modeled in Figure 4.10. In that figure, two cells must be repositioned to eliminate overlaps.
Figure 4.9: Area Model for Fraction Addition, 1/2 + 1/3 Example 4.2 Using
an area model, model the sum Solution 4.2
Figure 4.10: Area Model for Fraction Addition, 2/3 + 1/4 In Figures 4.9 and 4.10, the area model generated the greatest common divisor and least common denominator associated with the given fractions. The area model does not always produce this result. That is, although the area model always creates a common denominator, it does not necessarily create the least common denominator. That being the case, the region representing the solution will not be expressed in lowest terms. Figure 4.11 demonstrates this circumstance. After partitioning the unit square appropriately, a common divisor of 1/32 emerges. When 12 of these, representing 8/32 + 4/32, are repositioned as shown, we see that the fraction 12/32 may also be thought of as 3/8, or three rows in an eight row array. This model provides a useful representation for what we mean by a fraction reduced to lowest terms.
Figure 4.11: Area Model for Fraction Addition, 1/4 + 1/8 The fraction bar and area models may also be used to explain subtraction of fractions. Figure 4.12 models the operation 1/4 - 1/8. Procedurally, the only difference between the addition and subtraction models is the manner in which the cells are combined after being counted.
Figure 4.12: Area Model for Fraction Addition, 1/4 - 1/8 Example 4.3
Solution 4.3
Links National Library of Virtual Manipulatives for Interactive Mathematics
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