Counting by Fractions
Counting by fractions
Just as we count by ones, we can count by fractions: in steps of a half, a third, a quarter. From zero, counting by quarters goes 1/4, 2/4, 3/4, 4/4 — and 4/4 is one whole, so the next step is 5/4, past one. Counting this way shows that fractions are numbers you move through, not just parts of a single shape. The number line is the natural home for this: each fraction-sized step lands on the next mark, and Year 4 counts by fractions along the line, all the way past whole numbers into mixed numbers.
Locating a fraction
Every fraction has a definite place on the number line, found by counting fraction-sized steps from zero. To find 3/4, split the space from 0 to 1 into four equal parts and count three steps; to find 2/3, split into thirds and count two. Reading a marked point back into a fraction is the reverse: count the steps from 0 over the number that make a whole. Locating and reading fractions on the line, named in the descriptor, turns a fraction from an abstract pair of numbers into a spot you can point to, which is the key skill of this unit.
Mixed numbers
When a count of fractions passes a whole number, it becomes a mixed number: a whole number and a fraction together. One and a half is 1 1/2; two and a quarter is 2 1/4. The whole part counts the complete wholes, and the fraction part the leftover pieces. Mixed numbers are how we usually write amounts bigger than one in everyday life — one and a half cups, two and a quarter hours. Meeting mixed numerals is written into this descriptor, and they arise naturally the moment counting by fractions carries past one.
Improper fractions and mixed numbers
The same amount past one can be written two ways: as an improper fraction, where the top is bigger than the bottom, or as a mixed number. Five quarters, 5/4, is the same as 1 1/4, because four quarters make one whole and one quarter is left. Converting between them means asking how many wholes fit: how many groups of the bottom number go into the top. Seeing that 5/4 and 1 1/4 name the same point on the line connects the two notations, and it is a natural companion to counting by fractions, since counting produces improper fractions that mixed numbers tidy up.
Placing mixed numbers
A mixed number has its own place on the number line, between two whole numbers. 1 1/2 sits halfway between 1 and 2; 2 1/4 sits a quarter of the way from 2 to 3. To place it, find the whole part first, then step the fraction part past it. This shows mixed numbers are ordinary numbers with definite positions, not a separate kind of thing. Locating mixed numerals on the line, part of this descriptor, completes the picture: whole numbers, fractions and mixed numbers all live together on one line, each with its own spot.
Counts, mixed and improper
Pulling the unit together, counting by fractions runs along the number line and, past each whole, produces mixed numbers and improper fractions — two ways to write the same point. A table makes the links plain: counting by quarters reaches 5/4, which is 1 1/4. The number line ties it all together, giving every fraction and mixed number a place. With counting by fractions, fractions and mixed numbers located on the line, and improper and mixed forms connected, a child sees fractions as numbers among the whole numbers — the foundation for adding, comparing and working with them in later years.