Grades 3–5 · Mathematics
Why does a fraction depend on the whole?
A fraction describes equal parts of a named whole. One half means one of two equal parts of that whole. The fraction describes the relationship between a piece and its whole, but it does not give the piece's measured length without more information.
In the diagramThe fraction names a share of its own wholeWhole A: 8 units1/2One part of A: 4 unitsWhole B: 12 unitsOne part of B: 6 unitsDifferent whole lengths give different share lengths.
Strip A is eight length units long. Splitting it into two equal parts makes each half four units long. Strip B is twelve units long, so each of its equal halves is six units long. Both selected pieces are halves, yet the second piece has a greater measured length because its whole is longer.
The same idea applies to fourths. An eight-unit strip has two-unit fourths, while a twelve-unit strip has three-unit fourths. Within each strip, the shares are equal. Between different strips, the same fraction name need not mean the same length.
Comparisons as numbers on one scale require a fixed unit whole. Comparisons between physical pieces also need the sizes of their wholes. Naming the whole keeps these two kinds of comparison from being confused.
About this explanation
Starting knowledge: Recognize equal parts, read small whole-number lengths and use halves or fourths.
Sources
- Common Core State Standards Initiative — Grade 3: Number and Operations—Fractions