## Friday, September 7, 2007

### M5N4 (b) - Equivalent Fractions (2)

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.
a. Understand division of whole numbers can be represented as a fraction (a/b= a ÷ b).
b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one.

In the previous post, we looked at the idea of equivalent fractions. According to the GPS, students are expected to become aware that two fractions that look different may represent the same number, i.e., the concept of equivalent fractions, in Grade 4, while they are expected to understand how to create equivalent fractions by multiplying or dividing both the numerator and the denominator by the same number in Grade 5 (M5N4b). In this post, I would like to look at the reason why this procedure works.

In M5N4b, it is stated that this procedure works “because it is the same as multiplying or dividing by one.” This explanation makes perfect sense for us since we already know how to multiply or divide by fractions. For example,

However, this explanation is inappropriate for Grade 5 students because they have not studied how to multiply fractions by fractions. In Grade 5, students are expected to “model the multiplication and division of common fractions” (M5N4d). However, learning the procedures/algorithms for multiplying and dividing fractions is a Grade 6 expectation. Furthermore, even if students were to study the algorithms for multiplication and division of fractions in Grade 5, most likely they will be studying the procedure of creating equivalent fractions before they study multiplying and dividing fractions because they need to have this procedure to add and subtract fractions with unlike denominators. So, how can students understand that why this procedure for creating equivalent fractions work using only what they have already studied? Or, do they simply have to accept this procedure for now and justify it later?

Although there may be some topics in school mathematics where students may have to accept a formula or an algorithm just because teachers tell them it works, this is NOT such an occasion. Students do have something they have studied to justify this procedure. They key for understanding this procedure is M5N4a, which expands the meaning of fractions from simply a part of a whole or a collection of unit fractions (see my earlier post on the meaning of fractions). According to M5N4a, 5th grade students are expected to understand a fraction represents the answer for division of a whole number by another whole number. For example,

Moreover, in Grade 4, students are expected to “understand and explain the effect on the quotient of multiplying or dividing both the divisor and dividend by the same number” (M4N4d). Therefore, if you multiply or divide both 3 and 5 in the above example by the same number, the quotient does not change. For example,

But, this last expression, according to M5N4a, is the same as because a fraction represents the quotient of a whole number divided by another whole number. Or, in general,
, for any whole number .

The Georgia Performance Standards emphasizes that students should construct new mathematics understanding based on what they have previously studied. However, this is much easier said than done. Because we are already very familiar with mathematics students are still learning, we can easily overlook the fact that we slipped in something students have not studied yet. M5N4b is a very good illustration how difficult it is to create a rigorous, i.e., logically cohesive, curriculum.