Sometimes division leaves a remainder. If 14 ÷ 3, each group of 3 leaves 2 left over: 14 ÷ 3 = 4 R2 (four remainder two). The remainder is the amount left that's smaller than the divisor.
Use objects to physically divide and see what's left over. Draw pictures showing the remainder.
Ignoring the remainder; not understanding what it represents.
You already know division as equal sharing: if you have 12 cookies and 3 friends, each friend gets 4 cookies and nothing is left over. That's the clean, perfect case. But the real world is messier — most numbers don't divide evenly. That leftover amount is the remainder, and understanding it is just as important as finding the quotient.
Picture 14 apples being divided among 3 baskets. You put 4 in the first basket, 4 in the second, 4 in the third — that's 12 apples placed, and 2 are left. Two apples won't fill another full group of 3, so they stay as a remainder: 14 ÷ 3 = 4 remainder 2, written as 4 R2. Notice that the remainder is always *smaller than the divisor* — if it were 3 or more, you could fill another group.
One powerful check: multiply the quotient by the divisor, then add the remainder, and you should get the original number back. For 14 ÷ 3 = 4 R2: (4 × 3) + 2 = 12 + 2 = 14. ✓ This check catches mistakes because if the remainder is wrong, the total won't add up.
The trickiest part of remainders is knowing what to *do* with them in a word problem — and the answer depends on context. If 14 students need to fit into vans that hold 3, you need 5 vans (round up, because the 2 leftover students still need a ride). If you're cutting 3-foot pieces from a 14-foot rope, you get 4 pieces and 2 feet of scrap (keep the remainder as-is, or ignore it if the question asks only about complete pieces). The math gives you 4 R2 in both cases; it's the situation that tells you what to do with it. Always read the question to decide whether to round up, round down, or report the remainder.