The fraction a/b is equivalent to a divided by b. This is one of the most important conceptual connections in elementary mathematics. 3/4 means "3 divided into 4 equal parts," and each part is 0.75. This interpretation explains why fractions can represent quotients (sharing 3 pizzas among 4 people gives each person 3/4 of a pizza), connects fractions to decimals (just do the division), and makes the number line placement of fractions meaningful. It also explains why the fraction bar behaves like a division symbol.
Use fair-sharing problems: "Share 3 granola bars equally among 4 friends. How much does each person get?" Students draw and partition, discovering that each person gets 3/4 of a bar. Extend to other examples: 7 / 2 = 7/2 = 3 1/2. Connect to long division: dividing 3 by 4 on paper gives 0.75, and 3/4 = 0.75. Practice converting fractions to decimals via division.
You already know fractions as parts of a whole — 3/4 means 3 out of 4 equal pieces. And you know division as sharing — 12 ÷ 4 means splitting 12 into 4 equal groups. This topic fuses those two ideas: a/b and a ÷ b are the same thing. The fraction bar is a division symbol in disguise.
The clearest way to see this is through a sharing story. Suppose 3 friends share 3 granola bars equally. Each person gets 3 ÷ 3 = 1 bar. Now suppose 3 friends share 1 granola bar equally. Each gets 1 ÷ 3 = 1/3 of a bar. Now suppose 4 friends share 3 granola bars equally. Each person gets 3 ÷ 4 of a bar. How much is that? Draw it: give each friend 1/4 of each bar, across all 3 bars. Each friend collects 3 pieces of size 1/4, which totals 3/4 of a bar. So 3 ÷ 4 = 3/4. The division problem and the fraction are the same number.
This connection explains something that might have seemed strange: why does the fraction 3/4 produce the decimal 0.75? Because 3/4 means 3 ÷ 4, and if you carry out that long division, you get 0.75. The fraction is an instruction ("divide 3 by 4") and 0.75 is the result. Fractions and decimals aren't two different kinds of numbers — they're two ways of writing the same quantity. Whenever you want to convert a fraction to a decimal, just do the division.
The deeper implication is that any division problem can be written as a fraction, even when the result is less than 1. Before, you might have thought "5 ÷ 8 doesn't work because 8 doesn't go into 5." Now you know it does work — the answer is 5/8, which equals 0.625. Division always has an answer; it's just that sometimes the answer is a fraction. This unlocks dividing fractions later, where you'll need to trust that a ÷ b is always a valid number, no matter how a and b compare.