The Double Multiplicative Nature of Fraction or Ratio Equivalence

Most real-world numbers aren’t always so nice and neat, with wholenumber multiples. If, say, Plant A grew from 2 to 3 feet, and Plant B grew from 6 to 8 feet, then we would say that Plant A grew 1/2 of its original height, whereas Plant B only grew 1/3 of its original height. Such reasoning exemplifies multiplicative thinking and necessarily involves rational numbers.

Consider a final example. If you ask a rising 6th grader to compare 13/15 and 14/ 16, chances are that the student will say they are equal, because in both cases the numerator and denominator differ by 2. The student’s explanation might be, “I add the same number, 2, to the top number to get the bottom number, so they’re the same.” This is a testimony to ingrained additive thinking. Despite learning the equivalent fraction algorithm, most students leave elementary school unaware of the double multiplicative nature of equivalent fractions. Why? Because we don’t take the time to unpack it for them and then revisit it in multiple contexts!

Figure 0.4 illustrates both the between (or across ) ratio of 1 to 5 and the within (or downward ) ratio of 1 to 4 in two equivalent fractions. If this multiplicative nature of fractions were cultivated during the last three years of elementary school, then students wouldn’t think of comparing 13/15 and 14/16 additively. Multiplicative thinking underpins fractions, which in turn underpin the mathematics of ratios, rates, percents, proportions, linearity, and rational functions.