A fraction can describe a part of a group (set). For example, 2/3 of 6 objects means 4 objects. Comparing fractions with the same denominator (e.g., 2/4 vs. 3/4) is straightforward: the one with more parts is larger.
You've already worked with fractions of a set — finding 1/3 of 12 marbles by dividing the total into 3 equal groups and taking 1 of them. Now you're extending that to non-unit fractions: fractions where the numerator is greater than 1, like 2/3, 3/4, or 5/6. The process is the same — you still divide the set into equal groups first — but now you take more than one group.
Here's the two-step method. To find 2/3 of 12: first, the denominator (3) tells you how many equal groups to make — divide 12 into 3 groups of 4. Second, the numerator (2) tells you how many of those groups to take — take 2 groups of 4, which is 8. So 2/3 of 12 = 8. Notice that 1/3 of 12 is 4 (one group), and 2/3 is just double that — 8. The fraction acts like an instruction: split into this many groups, then select this many of them.
Comparing fractions with the same denominator is now straightforward because the denominator sets the "size" of each piece. If two fractions have the same denominator, every piece is the same size — the only difference is how many pieces you have. So 3/4 vs. 2/4: both use quarter-sized pieces; 3/4 just has one more piece, so it's larger. When denominators match, comparing numerators tells you everything.
This rule breaks down when denominators are different, which is why same-denominator comparison is the starting point rather than the whole story. But for now, the key idea to carry forward is that a fraction has two jobs: the denominator defines the unit (the size of each equal part), and the numerator counts how many of those units you have. Keeping those two roles distinct — size vs. count — is the foundation for all future fraction work.
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