3.NF.3.d

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.

4.NF.1 Creating equivalent fractions

4.NF.1 in the Common Core says, "Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions." When I first saw the expression (n x a)/(n x b), I thought that was odd. Usually I see math textbooks write this relationship as a/b = (a x n)/(b x n). Then I realized that the authors of the Common Core were trying to be consistent with the way they write multiplication expressions. "n x a" means "a is multiplied by n." Since we describe the process of creating equivalent fractions as "multiplying both the numerator and the denominator by the same non-zero number," it does make sense to write the expression as (n x a)/(n x b).

However, as I was thinking about how we might explain the process of equivalent fractions, I realized something else. One way, we can explain the process goes something like this - using 2/3 as an example. 2/3 is made up of 2 1/3-units, which is one of 3 equal parts of 1. So, if you use a diagram and a number line, it looks something like these:



Now, if we partition (split) each 1/3-unit into 4 equal pieces, we will have partitioned 1 into 12 equal parts, or 1/12-units. Pictorially, it will look like these:



Now, each 1/3-unit is made up of 4 1/12-units. So, 2 1/3-units are made up of 2 sets of 4 1/12-units. So, the number of 1/12-units in 2/3 is 2 x 4, and the number of 1/12-units in the whole is 3 x 4. So, 2/3 = (2 x 4)/(3 x 4), which is consistent with the conventional notation, a/b = an/bn.

I've been thinking about how we can use (n x a)/(n x b), but I haven't been successful, yet. I wonder if this is another instance where our language suggests the order of multiplication expression should be (group size) x (# of groups).