How can children's concepts of partitioning need to be refined to produce a concept of fractional parts?

Dividing up pizzas works.

(cut it into 8 parts or 6 parts or whatever)

So does dividing up money, especially based on a dollar.
(use ten dimes or four quarters or twenty nickles or one hundred pennies...)

why is it obvious that 3/4 x 8/5 = 6/5?

To understand how children's concepts of partitioning need to be refined to develop a concept of fractional parts, we need to explain the progression of their understanding.

Initially, children learn the concept of partitioning by dividing objects into equal parts. For example, they may divide a set of eight blocks into two sets of four. This initial understanding lays the foundation for the concept of fractions.

To develop a concept of fractional parts, children's understanding of partitioning needs to be refined in several ways:

1. Non-equal partitions: Children need to understand that objects can be divided into parts that are not necessarily equal. They need to grasp the idea that one part can be larger or smaller than the other.

2. Numerator and denominator: Children should learn that fractions consist of two numbers – a numerator and a denominator. The numerator represents the number of parts we have, while the denominator represents the total number of equal parts making up the whole. For example, in the fraction 3/4, the numerator is 3 (three parts) and the denominator is 4 (four equal parts).

3. Fraction as a number: Children need to recognize that fractions are numbers that can express quantities smaller than one whole. They should understand that fractions can be greater than one (improper fraction) or proper fractions (fractions smaller than one).

4. Equivalent fractions: Children should learn that fractions can be equivalent. For instance, 1/2 and 2/4 represent the same portion of a whole. They need to know that equivalent fractions have different numerators and denominators but represent the same amount.

5. Operations with fractions: As children refine their concept of fractional parts, they should learn how to perform basic operations with fractions, including addition, subtraction, multiplication, and division.

Overall, developing a concept of fractional parts requires refining children's understanding of partitioning by introducing the ideas of non-equal partitions, numerators and denominators, fractions as numbers, equivalent fractions, and operations with fractions. This progression helps children transition from the basic understanding of partitioning to a more sophisticated comprehension of fractional parts.