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We know that a fraction, as a "rational" number, is a ratio of two numbers. Note 2 In common usage, this ratio represents how many parts you have of a whole. But can a fraction have a different meaning?
There are actually two ways you might use fractional representation:
1. | One or more parts of a unit that has been divided into some number of equal-sized parts, which is a "part-whole" interpretation. For example, for the fraction 3/4, you might represent three slices of a pie that's been cut into four equal slices, or note that three out of every four balloons in a display are red. This is the interpretation most often used in the early and intermediate elementary grades. |
2. | One quantity in a whole compared to another quantity in a whole, which is a "part-part" interpretation. For example, to note that there are three red balloons for every four white balloons in a display, you would also use the fraction 3/4 (in this case, read as "three to four" rather than "three-fourths"). Note 3 |
Our standard rules for operations with fractions work perfectly with part-whole fractions, because the units are equivalent. These rules break down, however, when we look at part-part fractions.
For example, if Elizabeth has three red balloons and four green balloons, the ratio of red to green is 3/4. Suppose someone gives Elizabeth one more red balloon and two more green balloons (a ratio of 1/2). What is the ratio of red to green in Elizabeth's new collection? Let's explore this problem.
Students will be amazed to hear that, to answer this problem, they can "add" fractions the way they've always wanted to -- by simply adding the numerators and then adding the denominators!
3/4 + 1/2 = (3 + 1)/(4 + 2) = 4/6, or 2/3
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