### Background

A unit fraction is the base unit of any fraction and always has a numerator of 1; for example, ^{1}⁄_{4}, ^{1}⁄_{5}, ^{1}⁄_{23} are all unit
fractions. Every fraction can be decomposed into unit fractions. For example, ^{3}⁄_{4} is 3 one-fourth units (so one fourth is
the unit fraction and we are thinking about 3 of them). Partitioning a model involves determining and creating a unit fraction.

Consider the fraction one and three-fourths. This number can be decomposed using a unit fraction.

One and three-fourths

Seven one-fourth units

One and two-fourths can be composed using a unit fraction.

A student may say, “One whole is the same as 4 one-fourth units. I added another 2 one-fourth units to the whole to obtain 6 one-fourth units. So I can see that 6 one-fourth units is equal to one and two-fourths.

Use of unit fractions supports a deeper understanding of quantity. Notice that in the student dialogue above, early understanding of equivalency is being developed, "i.e., one and one-half is the same as six fourths". Counting by naming the unit fractions helps students to see the parts of the fraction when composing and decomposing. Notice that both counting unit fractions and composing and decomposing fractions are pre-cursors to addition and subtraction. For example, composing 6 one-fourth units is the same as adding 6 one-fourth units together to make one and one-half.