Some instructions and advice for teaching
The materials presented below are not addressed to children, but to the persons who want to teach children about fractions. In our opinion they present a bare minimum of knowledge and understanding required for teaching this subiect.
Children are familiar with the idea of fractions even before they start school and they may intuitively understand some simple fractions. After all they hear adults use fractions such as 1/2 a pie, 1/3 cup of milk, 3/4 of a mile, 1/2 dozen eggs and so on. However they do not understand fully the meaning of fractions.
Before children are taught about fractions they should be familiar at least with natural numbers or even whole numbers and their addition, subtraction, multiplication and to a certain extent division. For example they should realize that dividing 10 by 4 gives 2 with a reminder of 2. More formally
This approach, however, will not work with children. It is much too abstract. We have to explain them fractions by example using simple fraction interpretations. Moreover, this should be limited to the rational fractions only.
What are simple interpretations of a rational fraction?
The square in Fig. 1 is divided into four equally sized parts, consequently each of these parts is 1/4 of the square. If we treat the square as a unit or a whole, then any of its parts is just 1/4. Then the fraction 3/4 tells that we are considering three of these parts taken together, for example the three parts colored in blue. Exactly the same scheme applies to the rectangle and circle in Fig. 1. The rectangles in Fig. 2 illustrate the same kind of interpretation. Considering the parts that are not yellow, we deal with 1/3 and 2/6 whereas for the yellow parts with 2/3 and 4/6 respectively. Both pairs represent equivalent fractions describing or symbolizing the same rational number. Again, as for whole numbers, we can make any number of fractional representations of a given rational fraction. For example 1/3 can be also represented as 3/9 or 4/12. These examples show that simultaneous multiplication of the numerator and denominator of the fraction by the same non-zero number changes the fraction representation, but it does not change the rational number represented by the original fraction. This property is very important if we want to compare, add or subtract fractions.
Another possible interpretation of fractions can be illustrated with the twenty small squares in Fig. 1. If we consider columns, this set is divided into four equal groups of squares, five squares in each group (column). Thus, a single group constitutes 1/4 of the whole set, whereas all green squares constitute 3/4 of the set. Using this interpretation we can operate only with a limited number of fractions and the smallest fraction is one over the number of elements in the set. Consequently it is less usefull than the formerly discussed interpretation.
The easiest fractions to compare are fractions with the same denominator. Then the fraction with the bigger numerator is obviously the bigger fraction. But explaining it to children with a graphical representation like the one in Fig. 3 would not hurt.
Comparing 5/8 with 6/10 presents a more complicated problem and even adults have to think a moment to find the right answer. The graphical representation of the problem shown in the upper part of Fig. 4 gives the answer instantly, but it is not a practical method. The rational numbers 5/8 and 6/10 have many fractional representations. If we choose for both numbers their representations with the the same denominator, formally called the common denominator, the problem will be solved.
The simplest, but not optimal, method to find such denominator is to multiply the numerator and denominator of the first fraction by the denominator of the second fraction and the numerator and denominator of the second fraction by the denominator of the first fraction. In our example it replaces the original fractions 5/8 and 6/10 by 50/80 and 48/80 respectively. The new representation shows clearly that the first fraction is greater by 2/80. This difference that can be represented as 1/40 is shown in the lower part of Fig. 4.
Notice that the fraction 6/10 from our example can be reduced to its simplest equivalent form 3/5. Repetition of the procedure from the previous paragraph replaces the original fractions by 25/40 and 24/40. Thus, the simplification of 6/10 simplified in turn the comparison of both fractions showing right away that the difference between them is 1/40. In this case we have achieved an optimal simplification of our analysis. But for 5/6 and 7/10 the simplification of the original fractions is not possible and the simplest method of finding a common denominator is not good enough for optimization. This method will replace the original fractions by 50/60 and 42/60 showing that the first fraction is larger by 8/60. Further simplification replaces this difference by the equivalent fraction of 2/15.
The smallest common denominator for a set of fractions is called the least common denominator. Representation of these fractions in the forms with the least common denominator usually simplifies their comparison, addition and subtraction. Before a detailed discussion about finding the least commom denominators we shall talk about prime numbers. By definition every positive integer divisible only by one and itself (without any reminder) is a prime number. Thus 1, 2, 3, 5, 7, 11, 13, 17, 19 are prime numbers, whereas 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 are not. Moreover, from the definition of prime numbers follows an important property of non-prime positive integers. Namely each of them can be factored and represented as the product of at least two prime numbers different than 1. For example the non-prime numbers listed above can be presented as 2x2, 2x3, 2x2x2, 3x3, 2x7, 3x5,2x2x2x2, 2x3x3, 2x2x5 respectively.
Let us come back to our example with 5/6 and 7/10. The first denominator 6 = 2x3 and the second 10 = 2x5. Both of them have one common prime number 2, but the numbers 3 and 5 are not common. Multiplying this common prime number by the prime numbers from both denominators that are not common, we obtain 2x3x5 = 30 which is the least common denominator for this case. To find a new numerator for this denominator, the old numerator must be multiplied by all uncommon prime factors of the other denominator. Applying this rule to the current example our original fractions will be replaced by the equivalent fractions 25/30 and 21/30.
If instead of 10 the second denominator is 20 = 2x2x5 then we have one common pair of twos and another single 2 which is not common. Therefore multiplying the common 2 by the other noncommon prime factors we obtain the least common denominator 2x2x3x5 = 60.
The method described above always produces the least common denominator. First, the number created by the method is a common denominator because in the process of its creation we take the first denominator multiplied by the noncommon prime factors from the other denominator or vice-versa. Consequently we can divide it without any reminder either by the first or/and by the second denominator. To prove that it must be the least common denominator let us assume it is only a common denominator but not the least common denominator. If this is true, then we should be able to remove from it at least one prime factor and the reminder would still be a common denominator. But if we do so, from the construction of the original common denominator it follows that this remainder cannot be divided by at least one of the denominators of the original fractions. Thus, our assumption about the constructed common denominator as not the least common denominator was false.
In concluding this section we may say that usually the best method to handle comparison, addition or subtraction of fractions is to replace them, if possible, by their simplest representations and find for these representations the least common denominator. Replacement of a fraction by its simplest representation is typically called the reduction to the lowest term. Here again we may use the prime numbers technique. Consider the reduction of 18/30
Addition and subtraction of fractions
The most important methods useful for addition and subtraction were already discussed in the preceding section. Here you will find a few new definitions and some examples.
If the fractional value is smaller than 1 but greater than -1, the fraction is called a proper fraction. Otherwise the fraction is called an improper fraction. Thus for any proper fraction made of positive integers its denominator is always greater than the numerator. For the improper fraction of the same sort, the numerator is either greater than or equal to the denominator. The fractions 4/7, 7/9, 1/4, 2/4, -3/5 are proper fractions whereas 12/5, 5/5, 17/15, 10/5, -21/8 are improper fractions. Every improper fraction comprises non-zero number of units and a proper fraction. For example 12/5 = 2 2/5 or -21/8 = -2 5/8. The notation on the right sides of both examples is called the mixed numbers notation because it mixes whole numbers and fractions. The fractions used in this notation are usually proper fractions.
Using our knowledge about least common denominators let us try to add two proper fractions 2/3 and 1/4 and visualize this process with the help of our Fraction Addition and Subtraction Applet. After setting both fractions in the applet we have to determine the least common denominator. In this case the first denominator contains only one prime number 3 and the second contains the prime number 2 twice. The least common denominator is 12. The result of this choice is shown in Fig. 4 where the original fractions are visualized in the narrow rectangles and their equivalent representations pertaining to the least common denominator in the wide rectangles. The formal description of these rectangles appears on the little blackboards on the right. The result after addition is shown in Fig. 5. Formally this whole operation can be recodred as follows:
The subtraction operation of these two fractions can be recorded as:
and the visualization of the final result is shown in Fig. 6.
In the former example, the two added proper fractions produced the proper fraction too. But if we take 7/8 and 5/12 the result will be the improper fraction. Indeed, after concluding that the least common denominator in this case must be 24, we obtain
This addition procedure is illustrated in Fig. 7 and Fig. 8.
Recall that a mixed number contains a whole number and a fraction. Also recall that here we are only discussing here rational numbers. Thus it is always possible to convert a mixed number with an improper fraction into another mixed number with a proper fraction. In adding such mixed numbers we may first add their whole parts and then add the fractional parts utilizing the techniques described above. For example
A similar technique may be used to subtract fractions. In the following example we purpousely make the fractional part of the second mixed number greater than the fractional part of the first mixed number: