Division (II): With remainder

1Summary 重點摘要

7 ÷ 3 = 2 R1. Remainder < divisor.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Division is fair sharingDivision means sharing equally. Share 12 sweets among 3 people: each person gets 4, and 4 × 3 = 12 means the sweets are used up exactly. So 12 ÷ 3 = 4 with no remainder, and everyone gets the same amount. Equal sharing is the heart of division: the total, the number of parts and the size of each part are linked, and knowing any two gives the third. A common mistake is to assume division always comes out exact and to be lost when it does not; a remainder appears when it does not divide exactly, and it is a normal part of the answer. Understand equal sharing first, then division with remainders, in that order.
2. Part 2: Not exact means a remainderIf things cannot be shared equally, a remainder appears. Share 7 sweets among 3 people: giving 2 each uses 6 sweets, and the 1 left over is too few to go around again, so 7 ÷ 3 = 2 remainder 1. The quotient is 2 and the remainder is 1. After 2 sweets each, the last sweet belongs to no one in particular and stays behind. A common mistake is to give the last sweet to someone, spoiling the equality so that one person has 3 and the others 2. In division with remainders, the amount each person receives is fixed by the quotient; the remainder is not shared out, and the answer must state both clearly.
3. Part 3: The remainder is always smaller than the divisorThe remainder must be smaller than the divisor. For 7 ÷ 3 the remainder is 1, and 1 is less than 3. If 3 were left, every person could get one more sweet, so the quotient should be bigger. A remainder equal to or bigger than the divisor means the quotient is too small. For example, writing 17 ÷ 5 as 2 remainder 7 is wrong because 7 is greater than 5; the correct answer is 3 remainder 2. A common mistake is choosing a product too small when the tables are not fluent. After finding the quotient, always check whether the remainder is smaller than the divisor; this takes seconds and catches most errors, and once it becomes a habit, division with remainders rarely goes wrong.
4. Part 4: Use times tables to find the quotientUse times tables to find the quotient. For 25 ÷ 4, think of the 4 times table: 4 × 6 = 24 is not yet 25, and 4 × 7 = 28 is too big. Take the largest product that does not go past 25: the quotient is 6, and 25 − 24 = 1, so the answer is 6 remainder 1. Another example, 38 ÷ 5: 5 × 7 = 35 is not enough, 5 × 8 = 40 goes past, so the quotient is 7 with remainder 3. A common mistake is to pick a product past the dividend, taking the quotient of 25 ÷ 4 as 7. The rule is 'rather too small than too big': the product must not exceed the dividend, and what is left over is the remainder. Keep the direction clear and the quotient will not be wrong.
5. Part 5: How to checkCheck division by multiplying and then adding: quotient times divisor plus remainder should equal the dividend. For 25 ÷ 4 = 6 remainder 1: 6 × 4 + 1 = 24 + 1 = 25, which matches the number you started with, so the answer is correct. The order in the check matters too: multiply first, then add, never add first. A common mistake is to forget the remainder and compare only 6 × 4 = 24 with 25, wrongly concluding the answer is bad, or to remember the formula back to front as quotient plus divisor. The formula is quotient × divisor + remainder = dividend; jot it on scrap paper and follow it at every check, and nothing will be missed.
6. Part 6: Word problems depend on the questionIn word problems, what to do with the remainder depends on the question. For example, 29 sweets shared 4 per person: 29 ÷ 4 = 7 remainder 1, so 7 people get sweets and 1 sweet is left. If the question asks for the most people who can get sweets, take the quotient 7, because the leftover is not enough for one more person; if it asks how many more sweets are needed for one more person, that is 4 − 1 = 3. A common mistake is to ignore the question sentence and always answer with the quotient, or to round down wrongly in problems like ferrying people or packing boxes where everything must be dealt with. Before answering, decide: 'how many people or bags' uses the quotient; 'how many left' or 'how many more needed' uses the remainder or an extra calculation.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Division is fair sharing
• What cannot be shared is the remainder — always write it
• The remainder is always smaller than the divisor
• Use times tables to find the biggest quotient that fits
• Check: quotient × divisor + remainder = dividend