One can think of "a out of b" without actually removing the pieces. For example, one might think of the fraction of Xs in the diagram X X O O O as "two Xs out of five objects," in which case 2/5 of the objects are Xs. Can you apply this way of thinking to the fraction 6/5

To understand the fraction 6/5 using the concept of "a out of b," let's break it down.

The fraction 6/5 represents a ratio of two quantities: the numerator (6) and the denominator (5). In this case, it means we have six parts out of a total of five parts. However, this goes beyond the intuitive notion of a fraction as a part of a whole.

To make sense of 6/5 using the "a out of b" concept, you need to shift from thinking about physical objects to considering fractions as a general concept. Here's how you can do that:

1. Recognize that fractions can represent more than just parts out of a whole. They can also represent ratios, measures, or even values beyond a whole.

2. In the case of 6/5, you can think of it as having six parts out of five, regardless of the physical interpretation. This means comparing the numerator (6) to the denominator (5) to understand the relationship between the two quantities.

3. By recognizing that the denominator represents the reference value (or unit), you can interpret 6/5 as six units in relation to the reference value of five units. This implies that you have surpassed the original whole value by an extra unit.

In summary, when applying the "a out of b" concept to the fraction 6/5, you can think of it as having six units out of five units, which indicates having more than one whole. While this might not align with the traditional understanding of fractions, it allows you to think beyond the context of physical objects and consider fractions as general ratios or measures.