**3.NF.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.**

One of common misconceptions students (and adults) have about fractions is, "the closer the numerator is to the denominator, the larger the fraction." [Note, we are restricting our discussion to proper fractions.] They will say something like, 7/9 is greater than 2/5 because 7/9 is only 2 from the whole but 2/5 is 3 from the whole. What they do not realize is that they are confusing "closer to the denominator" with the idea, "closer to the whole." How close the numerator is to the denominator is indicated by subtraction, denominator - numerator. However, this difference simply indicates the number of unit fractions that are missing from the whole. How much is missing is determined not only by the number of unit fractions missing but also the size of those unit fractions. So, even though you may be missing 10 unit fractions from the whole, if each unit fraction is small, the total amount missing may be much less than 2 of larger unit fractions. So, 3/5, which is missing 2 1/5-units is missing more from the whole than 91/101, which are missing 10 1/101-units because 1/101-unit is much smaller than 1/5-units.

The 3.NF.3 standard states, "Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size." The intent of the standard, I believe, is for students to understand the idea that a (non-unit) fraction is composed of unit fractions. For example, 3/5 is made of 3 1/5-units. Thus, if you have two fractions with the same numerator, then those fractions are made of the same number of unit fractions. The sizes of those fractions depend on the sizes of the unit fractions. Since a unit fraction with a smaller denominator is greater than a unit fraction with a larger denominator, the fraction with the smaller denominator is made of the same number of larger units. Therefore, if the two fractions have the same numerators, the one with the smaller fraction is greater.

However, I wonder if comparison of fractions with the equal numerator also promotes some to develop the misconception I discussed above. For example, if you have 3/4 and 3/8, 3/4 is greater because the unit is greater. However, it is also the case that the numerator is closer to the denominator in 3/4 than in 3/8. I have no empirical evidence that this does happen, but it may be something 3rd grade teachers should keep in mind of this potential pitfall. The challenge is how can we help 3rd graders understand that the difference between the numerator and the denominator is not a reliable indicator for the size difference. There are some pairs of fractions where the one with the smaller difference is actually less than the one with the greater difference, for example, 1/2 vs. 4/6, 2/3 vs. 6/8, etc.. However, would that be enough? Are most 3rd graders developmentally ready to make sense of how deductive reasoning works. Do they understand that just because the less difference in 1/2 and 4/6 does not mean 1/2 is greater than 4/6, "the less difference" cannot be used as the rationale for concluding one fraction is greater than another? As 3rd grade teachers try to address this standard, it will be useful if they can make careful observations of their students and share their observations.

## 3 comments:

"the closer the numerator is to the denominator, the larger the fraction."..... I didn't know someone teaching or believing in the fraction comparison. If you have a resource (text books, especially) please share. In 3rd grade, they must understand "unit" fraction you mention along with equivalent fractions.

Perhaps careful choice of examples (especially the initial examples used when introducing comparisons) can prevent the misconception.

In other words, if few or none of the first comparison problems students are given are consistent with the misconception, the misconception will not develop in the first place.

I tend to believe that trying to prevent a misconception is a productive approach. Rather, we need to address (common) misconceptions explicitly - at the appropriate time. My preference would be not to discuss comparison of fractions with like numerators in Grade 3 since the focus of Grade 3 is for students to understand the unitary view of fractions - that is, non-unit fractions are collections of unit fractions.

Perhaps in Grade 4 or 5, students should consider the like numerator cases and then address the misconception explicitly.

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