Decomposing Fractions: Unit Fractions vs. Wholes
Decomposing Fractions: Unit Fractions vs. Wholes teaches Grade 3 students from Eureka Math that fractions greater than one can be broken apart in multiple equivalent ways. A fraction like 5/3 can be written as the sum of unit fractions (1/3 + 1/3 + 1/3 + 1/3 + 1/3), or as a whole and a unit fraction (3/3 + 2/3 = 1 + 2/3), or in other groupings. Recognizing that 3/3 = 1 whole is a key insight. This flexible decomposition builds the number sense needed for comparing fractions and transitioning to mixed numbers.
Key Concepts
A fraction greater than one can be decomposed in multiple ways. It can be written as a sum of its unit fractions, or it can be grouped and written as a sum of fractions that are each equivalent to one whole. For a whole number $W$ written as the fraction $\frac{a}{b}$: $$ \frac{a}{b} = \underbrace{\frac{1}{b} + \frac{1}{b} + \dots + \frac{1}{b}} {a \text{ times}} = \underbrace{\frac{b}{b} + \frac{b}{b} + \dots + \frac{b}{b}} {W \text{ times}} $$.
Common Questions
What is a unit fraction?
A unit fraction has 1 in the numerator: 1/2, 1/3, 1/4, etc. It represents one part of a whole divided into equal pieces.
How can you decompose 5/3 using unit fractions?
5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 — five unit fractions of 1/3.
How can you write 5/3 as a whole and a fraction?
5/3 = 3/3 + 2/3 = 1 + 2/3 = 1⅔.
What does 3/3 equal?
3/3 = 1 whole, because all 3 equal parts make up the entire unit.
Why is decomposing fractions useful in Grade 3?
It builds flexible thinking about fractional quantities, supporting fraction comparison, addition, and understanding of mixed numbers.
Can a fraction be decomposed more than one way?
Yes. 4/4 can be written as 1/4 + 3/4, or 2/4 + 2/4, or 1/4 + 1/4 + 2/4 — all are valid decompositions.