Common Core students partition things in third grade math, and they name the resulting pieces using fractions. Partitioning in this context means cuttings things into equal-sized pieces.
Children have lots of practice thinking about how to share something equally. When you cut a large cookie into five equal pieces so that you and your four friends each get the same amount, you make fifths. You call the size of each piece one-fifth, and you can write
The number language, the symbols, and thinking about number lines are all new for third graders. Partitioning into equal-sized pieces is familiar to them.
It's important that the pieces are equal‐sized, but they don't need to be the same shape. Each of the pieces in the square on the left in the figure is one‐fourth of the whole square, but that isn't true of the pieces of the square on the right.
The process of partitioning something into b same-sized pieces and then taking one of them gives you a fraction of the form
which is called a unit fraction. A unit in math is whatever you call one. This business of units is important. Misunderstandings of units result in all kinds of mathematical errors, so remember this basic principle: A unit is the thing that you count.
Consider the following conversation between two children after the younger one has made brownies with their mother.
Older child: How many brownies did you make?
Younger child: One big one. Mommy cut it up.
The older child is thinking that a brownie is the thing that you get when you cut up the pan. The younger child is thinking that a brownie is what you have before you cut it up. The younger child may be thinking that brownies work like cake — you make one cake and then cut it into slices. There is no good reason that in American, English‐speaking households, a cake is a unit that you partition into slices, while a brownie is a unit that comes from partitioning something larger.
Similar miscommunications about units are much less funny when there are consequences involved. When children first learn to count, one is the unit. When they count to five, they have counted five ones. Later, they learn to skip count. For example, they count by twos: two, four, six, eight, and so on. If you keep track on your fingers of the number of times you have counted by twos, your fingers are counting a different unit. Count to ten by twos and you'll have counted five times — that's five twos. Because the unit is the thing you count, this is an example of two as a unit.
When children study place value, they notice that 50 means five tens. You can count the tens in 50. Because the unit is the thing that you count, ten is a unit with a special role in the number system. Ten tens make one hundred, so one hundred is a unit. Place value depends on changing units — from ones to tens to hundreds and so on.
Now when students study fractions, they work with unit fractions. The unit fraction is a fraction with a 1 in the numerator, but the meaning of a unit fraction is that you can use them to count. Three‐fourths means three units of one‐fourth. You can count by fourths, just like you counted in kindergarten and first grade: one‐fourth, two‐fourths, three‐fourths, four‐fourths, five‐fourths, and so on.
This idea is emphasized in third grade. Fractions are built from an original unit by partitioning into equal‐sized pieces and then collecting some of these pieces. If you partition into b equal‐sized pieces, then collect a of them, it gives you the fraction
