Explaining and Justifying Fraction Operations by Raising Terms

Making Fraction Operations Easier to Master

Primary level workbooks and lessons show how raising terms leads to equivalent fractions and to explanations of how to add and subtract fractions and even mixed numbers involving different denominators. But raising terms can also make fractions comparison, multiplication and division easy to understand and explain.

In mathematics, we value the thought-based explanation or development of operations over giving them by rote. The explanation provides greater comprehension that some, if not all students, may appreciate in building their skills and knowledge. Mathematics when taught at a higher level is deeper when built on explanation of why methods work.

Comparison Example or Model

The question of which is more for a pair of fractions $\frac{11}4$ and $\frac{15}4$ with like denominators is easily answered as 15 is 4 more than 11. In contrast, the question of which is more for a pair of fractions $\frac{8}3$ and $\frac{5}2$ with nlike denominators is also easily answered by raising terms, so that both have a like denominator, smallest or not. Here 6 is a common denominator. So raising terms gives \[\frac{8}3 = \frac{2 \times 8}{2\times 3} = \frac{16}6\] while \[\frac{5}2 = \frac{3 \times 5}{3\times 3} = \frac{15}6\] So the difference, smaller than I thought is,$\frac{1}6$ and the first fraction $\frac{8}3$ is $\frac{1}6$ more than the second fraction $\frac{5}2$

The common cross-multiplication method of comparing fractions with unlike denominators follows from the model by using the product of the unlike denominators as a common denominator. In constrast, a smaller or least common denominator could be used. The latter would lead to smaller numbers but whether or not the latter would lead to a gain in efficiency in the comparision of fractions with small denominators and numerators is a question, not resolved here.

Multiplication by raising terms

Product of fractions may be computed by forming the product the numerators and denominators, respectively, to obtain the numerator and denominator of the product. Now a fifth of a multply of five, for example 15, is easily. For 15, a fifth is 3 and 4-fifths is twice as much, that is 12. For example

$\frac{4}{5}$ of $15 \times \frac{1}{7}$ is $12 \times \frac{1}{7}$ or $\frac{12}{7} = 1 \frac{5}{7}$

$\frac{1}{5}$ of $11$ is $\frac{1}{5}$ of $\frac{5 \times 11}{5}$ or $\frac{11}{5}$

Now 2-fifths would be twice as much.

$\frac{2}{5}$ of $11$ is 2 times as much or $\frac{2 \times 11}{5}= \frac{22}{5} $

In calculuting, one or more fifths of a fraction - the multiplicand say, we may raise terms in the latter so the numerator is a multiple of 5. For example,

\[\frac{3}{7} = \frac{5 \times 3 }{5 \times 7 } \]

by raising terms is equivalent to a fraction with a numerator equal a multiple of 5. Thus one fifth of the latter is \[\frac{1}{5} \mbox{ of } \frac{3}{7} = \frac{1 \times 3 }{5 \times 7 } \] and so fourth fifths would be four time more: \[\frac{4}{5} \mbox{ of } \frac{3}{7} = \frac{4 \times 3 }{5 \times 7 } \] Thus raising terms implies the multiply the tops, multiply the bottoms rule for multiplying factions: \[\frac{4}{5} \times \frac{3}{7} = \frac{4 \times 3 }{5 \times 7 } \]

Division by raising terms

The whole number 4 goes into 11, 2 wholes with a remainder of 3. The remainder is $\frac{3}{4}$ of 4. Thus the whole number 4 goes into 11, exactly

\[2 + \frac{3}{4}\]

times. That is,

\[11 \div 4 = 2 + \frac{3}{4} \]

Now 2 is $8 \times \frac{1}{4} = \frac{8}{4}$

Therefore

\[11 \div 4 = \frac{8}{4} + \frac{3}{4} = \frac{11}{4}\]

exactly.

Likewise 4 units of measure or counting goes into 11 units of measure or counting,

\[11 \div 4 = \frac{11}{4}\]

exactly.

Likewise $\frac{4}{5} = 4 \times \frac{1}{5}$ goes into $\frac{11}{5} = 11 \times \frac{1}{5}$ exactly

\[11 \div 4 = \frac{11}{4}\]

times. So in dividing fractions with common denomominators, the rule appears to be to ignore the common denominators: \[\frac{11}{5}\div \frac{4}{5} = \frac{11}{4}\]

As further example,

\[\frac{11}{99}\div \frac{4}{99} = \frac{11}{4}\]

The foregoing covers the like-denominator case. In general, we may raise terms to obtain a like denominator, as follows.

\[\frac{8}{3}\div \frac{5}{7} = \frac{8 \times 7}{3 \times 7}\div \frac{3 \times 5}{3 \times 7} = \frac{8 \times 7}{3 \times 5}\]

from the like denominator case. The result can be written in reciprocal mutliplication form:

\[\frac{8}{3}\div \frac{5}{7} = \frac{8}{3}\times \frac{7}{5} \]

where division by a fraction $\frac{5}{7}$ is obtained by multiplying by its reciprocal $ \frac{7}{5} $

Five Operations

In summary, the five operations of addition, subtraction, comparison, multiplication and division with fractions may be introduced and justified by raising terms.

Fraction Operations Efficiently

Two webpages

introduce this topic.

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