Year 5: Dividing in your head
Part of the Year 5 short division and remainders chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Dividing in your head
Many divisions can be done in your head if you use facts you already know. Use a times table and place value. You know that 56 ÷ 8 = 7. 560 is 10 times bigger than 56, so 560 ÷ 8 is 10 times bigger than 7: it is 70. And 5,600 ÷ 8 = 700.
| Division | Think | Answer |
|---|---|---|
| 56 ÷ 8 | 56 ÷ 8 = 7 | 7 |
| 560 ÷ 8 | 56 tens ÷ 8 = 7 tens | 70 |
| 5,600 ÷ 8 | 56 hundreds ÷ 8 = 7 hundreds | 700 |
| 5,600 ÷ 80 | take a zero off both: 560 ÷ 8 | 70 |
Multiplication and division are inverse (opposite) operations. They undo each other. If you know 7 × 80 = 560 then you also know 80 × 7 = 560, 560 ÷ 7 = 80 and 560 ÷ 80 = 7. These four facts are a fact family. The product (560) is always the number you divide.
Halving. Dividing by 2 is halving. Dividing by 4 is halving twice, and dividing by 8 is halving three times. Dividing by 5, 20 and 25. To divide by 5, double the number and divide by 10. To divide by 20, divide by 10 and halve. To divide by 25, multiply by 4 (double, double) and divide by 100. Splitting the divisor into factors. Dividing by 12 is the same as dividing by 3 and then by 4. Chunking. Split the number into pieces your divisor goes into easily, for example 156 = 120 + 36, then divide each piece by 6.
Key ideas
- Use a fact and place value: 42 ÷ 6 = 7 so 4,200 ÷ 6 = 700 and 4,200 ÷ 60 = 70.
- Fact family: 7 × 80 = 560, 80 × 7 = 560, 560 ÷ 7 = 80, 560 ÷ 80 = 7.
- ÷ 2 halve; ÷ 4 halve twice; ÷ 8 halve three times.
- ÷ 5: double, then ÷ 10. ÷ 20: ÷ 10, then halve. ÷ 25: double, double, then ÷ 100.
- Factors: ÷ 6 is ÷ 2 then ÷ 3. ÷ 12 is ÷ 3 then ÷ 4. Chunking: split into easy multiples of the divisor.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.560 ÷ 8: the digits are 56 ÷ 8
56 ÷ 8 = 7 because 8 × 7 = 56 - Use the fact on the leading digits. 560 is 56 tens. Share 56 tens into 8 equal groups.56 tens ÷ 8 = 7 tens
7 tens = 70 - Write the answer. 560 ÷ 8 = 70
Check. Multiply back: 70 × 8 = 560. ✓
Watch out. Count the zeros: 560 has 1 zero, so the answer has 1 zero after the 7.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.4,800 ÷ 80: the digits are 48 ÷ 8
48 ÷ 8 = 6 because 8 × 6 = 48 - Take 1 zero off both numbers. Dividing both numbers by 10 does not change the answer, and it makes the divisor a small number.4,800 ÷ 80 = 480 ÷ 8
- Use the fact on the leading digits. 480 is 48 tens. Share 48 tens into 8 equal groups.48 tens ÷ 8 = 6 tens
6 tens = 60 - Write the answer. 4,800 ÷ 80 = 60
Check. Multiply back: 60 × 80 = 4,800. ✓
Watch out. Count the zeros carefully: 4,800 has 2 zeros and 80 has 1 zero. Take the same number off both, then put back what is left.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Multiplying and dividing are inverse (opposite) operations. Use the multiplication fact to write the division fact.
- Look at the fact you are given. The three numbers in it are linked: two are factors and one is the product.7 × 80 = 560
- Write the division fact. Dividing undoes multiplying. 560 divided by one factor gives the other factor.560 ÷ 80 = 7
- Write the answer. The question asks for 560 ÷ 80.560 ÷ 80 = 7
Check. Multiply back: 7 × 80 = 560. ✓
Watch out. The product (560) is always the number you divide, never the number you divide by.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. A fact family has four facts that use the same three numbers: two multiplications and two divisions.
- Find the product. The product is the answer to the multiplication. It is the biggest number of the three.7 × 80 = 560
- Swap the factors. Multiplication can be done in either order, so there are two multiplication facts.7 × 80 = 560
80 × 7 = 560 - Write the two division facts. Start with the product. Divide it by each factor in turn.560 ÷ 7 = 80
560 ÷ 80 = 7
Check. Check one: 80 × 7 = 560. ✓
Watch out. The product starts both division facts. You cannot start with a factor.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 4 is the same as halving twice, because 4 = 2 × 2.
- Halve 864. Partition into place-value parts, halve each part, then add.864 = 800 + 60 + 4
half of each part: 400 + 30 + 2 = 432 - Halve 432 again. Partition into place-value parts, halve each part, then add.432 = 400 + 30 + 2
half of each part: 200 + 15 + 1 = 216 - Write the answer. 864 ÷ 4 = 216
Check. Double back: 216 × 4 = 864. ✓
Watch out. Halve twice, not once. Halving once is only ÷ 2.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. To divide by 5, double the number and then divide by 10. (5 is half of 10, so there are twice as many 5s as 10s.)
- Double the number. Doubling both numbers does not change the answer: n ÷ 5 is the same as 2n ÷ 10.345 = 300 + 40 + 5
double each part: 600 + 80 + 10 = 690 - Divide by 10. Move every digit one place to the right.690 ÷ 10 = 69
- Write the answer. 345 ÷ 5 = 69
Check. Multiply back: 69 × 5 = 345. ✓
Watch out. Double first, then divide by 10. Do not halve: that is for dividing by 20.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 20 is the same as dividing by 10 and then halving, because 20 = 10 × 2.
- Divide by 10. Move every digit one place to the right. (20 = 10 × 2.)640 ÷ 10 = 64
- Halve the answer. Now divide by the other factor, 2.64 = 60 + 4
half of each part: 30 + 2 = 32 - Write the answer. 640 ÷ 20 = 32
Check. Multiply back: 32 × 20 = 640. ✓
Watch out. Do both steps. Dividing by 10 on its own gives an answer that is twice too big.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. To divide by 25, multiply the number by 4 (double, double) and then divide by 100. (There are four 25s in 100.)
- Double the number. We multiply by 4 by doubling twice. (4 × 25 = 100.)650 = 600 + 50
double each part: 1,200 + 100 = 1,300 - Double again. Two doubles make × 4.1,300 = 1,000 + 300
double each part: 2,000 + 600 = 2,600 - Divide by 100. Multiplying both numbers by 4 does not change the answer: n ÷ 25 is the same as 4n ÷ 100. Move the digits two places to the right.2,600 ÷ 100 = 26
- Write the answer. 650 ÷ 25 = 26
Check. Multiply back: 26 × 25 = 650. ✓
Watch out. Remember there are two doubles (× 4), then divide by 100, not 10.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. 6 = 2 × 3, so dividing by 6 is the same as dividing by 2 and then by 3.
- Divide by 2. 6 = 2 × 3, so divide by 2 first.252 = 200 + 40 + 12
200 ÷ 2 = 100
40 ÷ 2 = 20
12 ÷ 2 = 6
100 + 20 + 6 = 126 - Divide the answer by 3. Now divide by the other factor.126 = 120 + 6
120 ÷ 3 = 40
6 ÷ 3 = 2
40 + 2 = 42 - Write the answer. 252 ÷ 6 = 42
Check. Multiply back: 42 × 6 = 252. ✓
Watch out. Divide the answer from step 1, not the original number again.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Split the number into chunks that are easy to divide (multiples of the divisor), divide each chunk, then add the answers.
- Split 156 into numbers that 6 divides into easily. Use multiples of 6, 60: the biggest chunk first.156 = 120 + 36
- Divide each chunk. Use a times table fact and place value.120 ÷ 6 = 20
36 ÷ 6 = 6 - Add the answers. The pieces of the answer add together.20 + 6 = 26
- Write the answer. 156 ÷ 6 = 26
Check. Multiply back: 26 × 6 = 156. ✓
Watch out. Check that the chunks add back to the original number.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Draw a bar model, choose the operation, then work it out.
- Draw a bar model. Each box is one pack of 25 pencils.
- Choose the operation. The pencils go into packs of 25, so we find how many 25s are in 450.450 ÷ 25
- Double the number. We multiply by 4 by doubling twice. (4 × 25 = 100.)450 = 400 + 50
double each part: 800 + 100 = 900 - Double again. Two doubles make × 4.900 = 900
double each part: 1,800 = 1,800 - Divide by 100. Multiplying both numbers by 4 does not change the answer: n ÷ 25 is the same as 4n ÷ 100. Move the digits two places to the right.1,800 ÷ 100 = 18
- Write the answer. Include the unit.18 packs
Check. Multiply back: 18 × 25 = 450. ✓
Watch out. Remember there are two doubles (× 4), then divide by 100, not 10.
Watch out
Count the zeros with care. 4,200 ÷ 60 is 70, not 700. Take the same number of zeros off both numbers (one each), then use the fact 42 ÷ 6 = 7 and put back the zero that is left. Also do not mix up the inverse facts: the product, 560, is the number you divide. 80 ÷ 560 is not the same as 560 ÷ 80.Exercise 5B Dividing in your head
Non-calculatorKey words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 9 by to get 36?9 × ? = 36
- Count up in 9s. Say the 9 times table until you reach 36.9, 18, 27, 36
- Write the division. You said 4 numbers, so 9 × 4 = 36.36 ÷ 9 = 4
Check. 4 × 9 = 36. ✓
Watch out. Check your answer by multiplying: it should give 36.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 8 by to get 72?8 × ? = 72
- Count up in 8s. Say the 8 times table until you reach 72.8, 16, 24, 32, 40, 48, 56, 64, 72
- Write the division. You said 9 numbers, so 8 × 9 = 72.72 ÷ 8 = 9
Check. 9 × 8 = 72. ✓
Watch out. Check your answer by multiplying: it should give 72.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 7 by to get 63?7 × ? = 63
- Count up in 7s. Say the 7 times table until you reach 63.7, 14, 21, 28, 35, 42, 49, 56, 63
- Write the division. You said 9 numbers, so 7 × 9 = 63.63 ÷ 7 = 9
Check. 9 × 7 = 63. ✓
Watch out. Check your answer by multiplying: it should give 63.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 12 by to get 48?12 × ? = 48
- Count up in 12s. Say the 12 times table until you reach 48.12, 24, 36, 48
- Write the division. You said 4 numbers, so 12 × 4 = 48.48 ÷ 12 = 4
Check. 4 × 12 = 48. ✓
Watch out. Check your answer by multiplying: it should give 48.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 11 by to get 121?11 × ? = 121
- Count up in 11s. Say the 11 times table until you reach 121.11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121
- Write the division. You said 11 numbers, so 11 × 11 = 121.121 ÷ 11 = 11
Check. 11 × 11 = 121. ✓
Watch out. Check your answer by multiplying: it should give 121.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 12 by to get 132?12 × ? = 132
- Count up in 12s. Say the 12 times table until you reach 132.12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132
- Write the division. You said 11 numbers, so 12 × 11 = 132.132 ÷ 12 = 11
Check. 11 × 12 = 132. ✓
Watch out. Check your answer by multiplying: it should give 132.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 6 by to get 54?6 × ? = 54
- Count up in 6s. Say the 6 times table until you reach 54.6, 12, 18, 24, 30, 36, 42, 48, 54
- Write the division. You said 9 numbers, so 6 × 9 = 54.54 ÷ 6 = 9
Check. 9 × 6 = 54. ✓
Watch out. Check your answer by multiplying: it should give 54.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Use the times table. Division is the opposite of multiplication.
- Think of the times table. Division is the opposite of multiplication. Ask: what do I multiply 9 by to get 81?9 × ? = 81
- Count up in 9s. Say the 9 times table until you reach 81.9, 18, 27, 36, 45, 54, 63, 72, 81
- Write the division. You said 9 numbers, so 9 × 9 = 81.81 ÷ 9 = 9
Check. 9 × 9 = 81. ✓
Watch out. Check your answer by multiplying: it should give 81.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.3,600 ÷ 9: the digits are 36 ÷ 9
36 ÷ 9 = 4 because 9 × 4 = 36 - Use the fact on the leading digits. 3,600 is 36 hundreds. Share 36 hundreds into 9 equal groups.36 hundreds ÷ 9 = 4 hundreds
4 hundreds = 400 - Write the answer. 3,600 ÷ 9 = 400
Check. Multiply back: 400 × 9 = 3,600. ✓
Watch out. Count the zeros: 3,600 has 2 zeros, so the answer has 2 zeros after the 4.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.280 ÷ 7: the digits are 28 ÷ 7
28 ÷ 7 = 4 because 7 × 4 = 28 - Use the fact on the leading digits. 280 is 28 tens. Share 28 tens into 7 equal groups.28 tens ÷ 7 = 4 tens
4 tens = 40 - Write the answer. 280 ÷ 7 = 40
Check. Multiply back: 40 × 7 = 280. ✓
Watch out. Count the zeros: 280 has 1 zero, so the answer has 1 zero after the 4.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.4,500 ÷ 5: the digits are 45 ÷ 5
45 ÷ 5 = 9 because 5 × 9 = 45 - Use the fact on the leading digits. 4,500 is 45 hundreds. Share 45 hundreds into 5 equal groups.45 hundreds ÷ 5 = 9 hundreds
9 hundreds = 900 - Write the answer. 4,500 ÷ 5 = 900
Check. Multiply back: 900 × 5 = 4,500. ✓
Watch out. Count the zeros: 4,500 has 2 zeros, so the answer has 2 zeros after the 9.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.810 ÷ 9: the digits are 81 ÷ 9
81 ÷ 9 = 9 because 9 × 9 = 81 - Use the fact on the leading digits. 810 is 81 tens. Share 81 tens into 9 equal groups.81 tens ÷ 9 = 9 tens
9 tens = 90 - Write the answer. 810 ÷ 9 = 90
Check. Multiply back: 90 × 9 = 810. ✓
Watch out. Count the zeros: 810 has 1 zero, so the answer has 1 zero after the 9.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.4,200 ÷ 60: the digits are 42 ÷ 6
42 ÷ 6 = 7 because 6 × 7 = 42 - Take 1 zero off both numbers. Dividing both numbers by 10 does not change the answer, and it makes the divisor a small number.4,200 ÷ 60 = 420 ÷ 6
- Use the fact on the leading digits. 420 is 42 tens. Share 42 tens into 6 equal groups.42 tens ÷ 6 = 7 tens
7 tens = 70 - Write the answer. 4,200 ÷ 60 = 70
Check. Multiply back: 70 × 60 = 4,200. ✓
Watch out. Count the zeros carefully: 4,200 has 2 zeros and 60 has 1 zero. Take the same number off both, then put back what is left.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.3,600 ÷ 90: the digits are 36 ÷ 9
36 ÷ 9 = 4 because 9 × 4 = 36 - Take 1 zero off both numbers. Dividing both numbers by 10 does not change the answer, and it makes the divisor a small number.3,600 ÷ 90 = 360 ÷ 9
- Use the fact on the leading digits. 360 is 36 tens. Share 36 tens into 9 equal groups.36 tens ÷ 9 = 4 tens
4 tens = 40 - Write the answer. 3,600 ÷ 90 = 40
Check. Multiply back: 40 × 90 = 3,600. ✓
Watch out. Count the zeros carefully: 3,600 has 2 zeros and 90 has 1 zero. Take the same number off both, then put back what is left.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.560 ÷ 70: the digits are 56 ÷ 7
56 ÷ 7 = 8 because 7 × 8 = 56 - Take 1 zero off both numbers. Dividing both numbers by 10 does not change the answer, and it makes the divisor a small number.560 ÷ 70 = 56 ÷ 7
- Use the fact. After taking the zeros away there is nothing else to put back.56 ÷ 7 = 8
- Write the answer. 560 ÷ 70 = 8
Check. Multiply back: 8 × 70 = 560. ✓
Watch out. Count the zeros carefully: 560 has 1 zero and 70 has 1 zero. Take the same number off both, then put back what is left.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.8,100 ÷ 900: the digits are 81 ÷ 9
81 ÷ 9 = 9 because 9 × 9 = 81 - Take 2 zeros off both numbers. Dividing both numbers by 100 does not change the answer, and it makes the divisor a small number.8,100 ÷ 900 = 81 ÷ 9
- Use the fact. After taking the zeros away there is nothing else to put back.81 ÷ 9 = 9
- Write the answer. 8,100 ÷ 900 = 9
Check. Multiply back: 9 × 900 = 8,100. ✓
Watch out. Count the zeros carefully: 8,100 has 2 zeros and 900 has 2 zeros. Take the same number off both, then put back what is left.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Multiplying and dividing are inverse (opposite) operations. Use the multiplication fact to write the division fact.
- Look at the fact you are given. The three numbers in it are linked: two are factors and one is the product.6 × 40 = 240
- Write the division fact. Dividing undoes multiplying. 240 divided by one factor gives the other factor.240 ÷ 6 = 40
- Write the answer. The question asks for 240 ÷ 6.240 ÷ 6 = 40
Check. Multiply back: 40 × 6 = 240. ✓
Watch out. The product (240) is always the number you divide, never the number you divide by.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Multiplying and dividing are inverse (opposite) operations. Use the multiplication fact to write the division fact.
- Look at the fact you are given. The three numbers in it are linked: two are factors and one is the product.6 × 40 = 240
- Write the division fact. Dividing undoes multiplying. 240 divided by one factor gives the other factor.240 ÷ 40 = 6
- Write the answer. The question asks for 240 ÷ 40.240 ÷ 40 = 6
Check. Multiply back: 6 × 40 = 240. ✓
Watch out. The product (240) is always the number you divide, never the number you divide by.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Use a times table fact. Work out the fact with the zeros hidden, then put the right number of zeros back.
- Spot the times table fact. Cover the zeros. The digits that are left make a fact you know.2,400 ÷ 6: the digits are 24 ÷ 6
24 ÷ 6 = 4 because 6 × 4 = 24 - Use the fact on the leading digits. 2,400 is 24 hundreds. Share 24 hundreds into 6 equal groups.24 hundreds ÷ 6 = 4 hundreds
4 hundreds = 400 - Write the answer. 2,400 ÷ 6 = 400
Check. Multiply back: 400 × 6 = 2,400. ✓
Watch out. Count the zeros: 2,400 has 2 zeros, so the answer has 2 zeros after the 4.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. A fact family has four facts that use the same three numbers: two multiplications and two divisions.
- Find the product. The product is the answer to the multiplication. It is the biggest number of the three.9 × 6 = 54
- Swap the factors. Multiplication can be done in either order, so there are two multiplication facts.9 × 6 = 54
6 × 9 = 54 - Write the two division facts. Start with the product. Divide it by each factor in turn.54 ÷ 9 = 6
54 ÷ 6 = 9
Check. Check one: 6 × 9 = 54. ✓
Watch out. The product starts both division facts. You cannot start with a factor.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. A fact family has four facts that use the same three numbers: two multiplications and two divisions.
- Find the product. The product is the answer to the multiplication. It is the biggest number of the three.8 × 70 = 560
- Swap the factors. Multiplication can be done in either order, so there are two multiplication facts.8 × 70 = 560
70 × 8 = 560 - Write the two division facts. Start with the product. Divide it by each factor in turn.560 ÷ 8 = 70
560 ÷ 70 = 8
Check. Check one: 70 × 8 = 560. ✓
Watch out. The product starts both division facts. You cannot start with a factor.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Division and multiplication undo each other. Use the opposite operation to find the missing number.
- Use the opposite operation. The missing number was divided by 7 to make 12. To undo the division, multiply.12 × 7
- Work it out. Use a times table fact or place value.12 × 7 = 84
Check. 84 ÷ 7 = 12. ✓
Watch out. Put your answer back in the box and read the whole sentence.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Division and multiplication undo each other. Use the opposite operation to find the missing number.
- Use the opposite operation. The missing number was divided by 9 to make 8. To undo the division, multiply.8 × 9
- Work it out. Use a times table fact or place value.8 × 9 = 72
Check. 72 ÷ 9 = 8. ✓
Watch out. Put your answer back in the box and read the whole sentence.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Write the matching multiplication, then use the fact family to find the missing number.
- Think about the fact family. 72 is the product. It is shared into groups of the missing size to make 9 groups.9 × ? = 72
- Divide the product by the number you know. The missing factor is the product divided by the factor we know.72 ÷ 9 = 8
Check. 72 ÷ 8 = 9. ✓
Watch out. The missing number can be the divisor: the number you divide by.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Write the matching multiplication, then use the fact family to find the missing number.
- Think about the fact family. 55 is the product. It is shared into groups of the missing size to make 11 groups.11 × ? = 55
- Divide the product by the number you know. The missing factor is the product divided by the factor we know.55 ÷ 11 = 5
Check. 55 ÷ 5 = 11. ✓
Watch out. The missing number can be the divisor: the number you divide by.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 2 is the same as halving.
- Halve 148. Partition into place-value parts, halve each part, then add.148 = 100 + 40 + 8
half of each part: 50 + 20 + 4 = 74 - Write the answer. 148 ÷ 2 = 74
Check. Double back: 74 × 2 = 148. ✓
Watch out. Halve each place-value part and add the halves back together.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 2 is the same as halving.
- Halve 356. Partition into place-value parts, halve each part, then add.356 = 300 + 50 + 6
half of each part: 150 + 25 + 3 = 178 - Write the answer. 356 ÷ 2 = 178
Check. Double back: 178 × 2 = 356. ✓
Watch out. Halve each place-value part and add the halves back together.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 2 is the same as halving.
- Halve 762. Partition into place-value parts, halve each part, then add.762 = 700 + 60 + 2
half of each part: 350 + 30 + 1 = 381 - Write the answer. 762 ÷ 2 = 381
Check. Double back: 381 × 2 = 762. ✓
Watch out. Halve each place-value part and add the halves back together.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 2 is the same as halving.
- Halve 1,584. Partition into place-value parts, halve each part, then add.1,584 = 1,000 + 500 + 80 + 4
half of each part: 500 + 250 + 40 + 2 = 792 - Write the answer. 1,584 ÷ 2 = 792
Check. Double back: 792 × 2 = 1,584. ✓
Watch out. Halve each place-value part and add the halves back together.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 4 is the same as halving twice, because 4 = 2 × 2.
- Halve 316. Partition into place-value parts, halve each part, then add.316 = 300 + 10 + 6
half of each part: 150 + 5 + 3 = 158 - Halve 158 again. Partition into place-value parts, halve each part, then add.158 = 100 + 50 + 8
half of each part: 50 + 25 + 4 = 79 - Write the answer. 316 ÷ 4 = 79
Check. Double back: 79 × 4 = 316. ✓
Watch out. Halve twice, not once. Halving once is only ÷ 2.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 4 is the same as halving twice, because 4 = 2 × 2.
- Halve 528. Partition into place-value parts, halve each part, then add.528 = 500 + 20 + 8
half of each part: 250 + 10 + 4 = 264 - Halve 264 again. Partition into place-value parts, halve each part, then add.264 = 200 + 60 + 4
half of each part: 100 + 30 + 2 = 132 - Write the answer. 528 ÷ 4 = 132
Check. Double back: 132 × 4 = 528. ✓
Watch out. Halve twice, not once. Halving once is only ÷ 2.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 8 is the same as halving three times, because 8 = 2 × 2 × 2.
- Halve 456. Partition into place-value parts, halve each part, then add.456 = 400 + 50 + 6
half of each part: 200 + 25 + 3 = 228 - Halve 228 again. Partition into place-value parts, halve each part, then add.228 = 200 + 20 + 8
half of each part: 100 + 10 + 4 = 114 - Halve 114 again. Partition into place-value parts, halve each part, then add.114 = 100 + 10 + 4
half of each part: 50 + 5 + 2 = 57 - Write the answer. 456 ÷ 8 = 57
Check. Double back: 57 × 8 = 456. ✓
Watch out. Halve three times, not once. Halving once is only ÷ 2.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 8 is the same as halving three times, because 8 = 2 × 2 × 2.
- Halve 1,208. Partition into place-value parts, halve each part, then add.1,208 = 1,000 + 200 + 8
half of each part: 500 + 100 + 4 = 604 - Halve 604 again. Partition into place-value parts, halve each part, then add.604 = 600 + 4
half of each part: 300 + 2 = 302 - Halve 302 again. Partition into place-value parts, halve each part, then add.302 = 300 + 2
half of each part: 150 + 1 = 151 - Write the answer. 1,208 ÷ 8 = 151
Check. Double back: 151 × 8 = 1,208. ✓
Watch out. Halve three times, not once. Halving once is only ÷ 2.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. To divide by 5, double the number and then divide by 10. (5 is half of 10, so there are twice as many 5s as 10s.)
- Double the number. Doubling both numbers does not change the answer: n ÷ 5 is the same as 2n ÷ 10.235 = 200 + 30 + 5
double each part: 400 + 60 + 10 = 470 - Divide by 10. Move every digit one place to the right.470 ÷ 10 = 47
- Write the answer. 235 ÷ 5 = 47
Check. Multiply back: 47 × 5 = 235. ✓
Watch out. Double first, then divide by 10. Do not halve: that is for dividing by 20.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Dividing by 20 is the same as dividing by 10 and then halving, because 20 = 10 × 2.
- Divide by 10. Move every digit one place to the right. (20 = 10 × 2.)460 ÷ 10 = 46
- Halve the answer. Now divide by the other factor, 2.46 = 40 + 6
half of each part: 20 + 3 = 23 - Write the answer. 460 ÷ 20 = 23
Check. Multiply back: 23 × 20 = 460. ✓
Watch out. Do both steps. Dividing by 10 on its own gives an answer that is twice too big.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. To divide by 25, multiply the number by 4 (double, double) and then divide by 100. (There are four 25s in 100.)
- Double the number. We multiply by 4 by doubling twice. (4 × 25 = 100.)375 = 300 + 70 + 5
double each part: 600 + 140 + 10 = 750 - Double again. Two doubles make × 4.750 = 700 + 50
double each part: 1,400 + 100 = 1,500 - Divide by 100. Multiplying both numbers by 4 does not change the answer: n ÷ 25 is the same as 4n ÷ 100. Move the digits two places to the right.1,500 ÷ 100 = 15
- Write the answer. 375 ÷ 25 = 15
Check. Multiply back: 15 × 25 = 375. ✓
Watch out. Remember there are two doubles (× 4), then divide by 100, not 10.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. To divide by 25, multiply the number by 4 (double, double) and then divide by 100. (There are four 25s in 100.)
- Double the number. We multiply by 4 by doubling twice. (4 × 25 = 100.)1,200 = 1,000 + 200
double each part: 2,000 + 400 = 2,400 - Double again. Two doubles make × 4.2,400 = 2,000 + 400
double each part: 4,000 + 800 = 4,800 - Divide by 100. Multiplying both numbers by 4 does not change the answer: n ÷ 25 is the same as 4n ÷ 100. Move the digits two places to the right.4,800 ÷ 100 = 48
- Write the answer. 1,200 ÷ 25 = 48
Check. Multiply back: 48 × 25 = 1,200. ✓
Watch out. Remember there are two doubles (× 4), then divide by 100, not 10.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. 6 = 2 × 3, so dividing by 6 is the same as dividing by 2 and then by 3.
- Divide by 2. 6 = 2 × 3, so divide by 2 first.324 = 200 + 120 + 4
200 ÷ 2 = 100
120 ÷ 2 = 60
4 ÷ 2 = 2
100 + 60 + 2 = 162 - Divide the answer by 3. Now divide by the other factor.162 = 150 + 12
150 ÷ 3 = 50
12 ÷ 3 = 4
50 + 4 = 54 - Write the answer. 324 ÷ 6 = 54
Check. Multiply back: 54 × 6 = 324. ✓
Watch out. Divide the answer from step 1, not the original number again.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. 12 = 3 × 4, so dividing by 12 is the same as dividing by 3 and then by 4.
- Divide by 3. 12 = 3 × 4, so divide by 3 first.252 = 240 + 12
240 ÷ 3 = 80
12 ÷ 3 = 4
80 + 4 = 84 - Divide the answer by 4. Now divide by the other factor.84 = 80 + 4
80 ÷ 4 = 20
4 ÷ 4 = 1
20 + 1 = 21 - Write the answer. 252 ÷ 12 = 21
Check. Multiply back: 21 × 12 = 252. ✓
Watch out. Divide the answer from step 1, not the original number again.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. 15 = 3 × 5, so dividing by 15 is the same as dividing by 3 and then by 5.
- Divide by 3. 15 = 3 × 5, so divide by 3 first.480 = 300 + 180
300 ÷ 3 = 100
180 ÷ 3 = 60
100 + 60 = 160 - Divide the answer by 5. Now divide by the other factor.160 = 150 + 10
150 ÷ 5 = 30
10 ÷ 5 = 2
30 + 2 = 32 - Write the answer. 480 ÷ 15 = 32
Check. Multiply back: 32 × 15 = 480. ✓
Watch out. Divide the answer from step 1, not the original number again.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Split the number into chunks that are easy to divide (multiples of the divisor), divide each chunk, then add the answers.
- Split 168 into numbers that 7 divides into easily. Use multiples of 7, 70: the biggest chunk first.168 = 140 + 28
- Divide each chunk. Use a times table fact and place value.140 ÷ 7 = 20
28 ÷ 7 = 4 - Add the answers. The pieces of the answer add together.20 + 4 = 24
- Write the answer. 168 ÷ 7 = 24
Check. Multiply back: 24 × 7 = 168. ✓
Watch out. Check that the chunks add back to the original number.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Split the number into chunks that are easy to divide (multiples of the divisor), divide each chunk, then add the answers.
- Split 1,140 into numbers that 6 divides into easily. Use multiples of 6, 60, 600: the biggest chunk first.1,140 = 600 + 540
- Divide each chunk. Use a times table fact and place value.600 ÷ 6 = 100
540 ÷ 6 = 90 - Add the answers. The pieces of the answer add together.100 + 90 = 190
- Write the answer. 1,140 ÷ 6 = 190
Check. Multiply back: 190 × 6 = 1,140. ✓
Watch out. Check that the chunks add back to the original number.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Split the number into chunks that are easy to divide (multiples of the divisor), divide each chunk, then add the answers.
- Split 3,150 into numbers that 5 divides into easily. Use multiples of 5, 50, 500: the biggest chunk first.3,150 = 3,000 + 150
- Divide each chunk. Use a times table fact and place value.3,000 ÷ 5 = 600
150 ÷ 5 = 30 - Add the answers. The pieces of the answer add together.600 + 30 = 630
- Write the answer. 3,150 ÷ 5 = 630
Check. Multiply back: 630 × 5 = 3,150. ✓
Watch out. Check that the chunks add back to the original number.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Draw a bar model, choose the operation, then work it out.
- Draw a bar model. Each box is one box of 25 cards.
- Choose the operation. The cards go into boxes of 25, so we find how many 25s are in 600.600 ÷ 25
- Double the number. We multiply by 4 by doubling twice. (4 × 25 = 100.)600 = 600
double each part: 1,200 = 1,200 - Double again. Two doubles make × 4.1,200 = 1,000 + 200
double each part: 2,000 + 400 = 2,400 - Divide by 100. Multiplying both numbers by 4 does not change the answer: n ÷ 25 is the same as 4n ÷ 100. Move the digits two places to the right.2,400 ÷ 100 = 24
- Write the answer. Include the unit.24 boxes
Check. Multiply back: 24 × 25 = 600. ✓
Watch out. Remember there are two doubles (× 4), then divide by 100, not 10.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Draw a bar model, choose the operation, then work it out.
- Draw a bar model. Each box is one bag of 20 rolls.
- Choose the operation. The rolls go into bags of 20, so we find how many 20s are in 960.960 ÷ 20
- Divide by 10. Move every digit one place to the right. (20 = 10 × 2.)960 ÷ 10 = 96
- Halve the answer. Now divide by the other factor, 2.96 = 90 + 6
half of each part: 45 + 3 = 48 - Write the answer. Include the unit.48 bags
Check. Multiply back: 48 × 20 = 960. ✓
Watch out. Do both steps. Dividing by 10 on its own gives an answer that is twice too big.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.56 ÷ 8 = 7. But 560 is 10 times bigger than 56, so the answer is 10 times bigger.
560 ÷ 8 = 70 - Compare with the answer in the question. Ben wrote 7, which is the answer to 56 ÷ 8, not 560 ÷ 8.
- Find the mistake. Ben used the fact but forgot to put the zero back. 560 is 56 tens, so the answer is 7 tens.
- Write the correct answer. The correct answer is 70.
Check. 70 × 8 = 560. ✓
Watch out. After using a fact, put the zeros back. Check by multiplying: 7 × 8 = 56, not 560.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To halve is to divide by 2. To double is to multiply by 2.
Method. To prove it, work out both ways for several numbers and compare. Then explain why it always works.
- Divide by 4 and halve twice. Work out both ways for each number.
Number Divide by 4 Halve, then halve again Same? 36 36 ÷ 4 = 9 36 ÷ 2 = 18, 18 ÷ 2 = 9 yes 72 72 ÷ 4 = 18 72 ÷ 2 = 36, 36 ÷ 2 = 18 yes 124 124 ÷ 4 = 31 124 ÷ 2 = 62, 62 ÷ 2 = 31 yes - Compare. Every row gives the same answer.All the pairs match.
- Explain why. Halving means dividing by 2. Dividing by 2 and then by 2 again is the same as dividing by 2 × 2.2 × 2 = 4, so ÷ 2 ÷ 2 is the same as ÷ 4.
Check. Try one more: 108 ÷ 4 = 27 and half of half of 108 = 27. ✓
Watch out. “Prove it” needs working AND a sentence of explanation. Examples alone are not enough.
Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Dividing by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times smaller.
Method. Work out the answer to every one. The odd one out has a different answer from the others.
- Work out each one. Use place value or a known fact.
Calculation Answer 560 ÷ 8 70 490 ÷ 7 70 4,200 ÷ 60 70 3,600 ÷ 60 60 - Find the odd one out. Three have the same answer, 70. One is different.3,600 ÷ 60 = 60
- Write the answer. The odd one out is 3,600 ÷ 60.
Check. The other three all equal 70. ✓
Watch out. Work out all of them. Do not decide from just the first two.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. Work backwards. Undo each step in the opposite order, using the opposite (inverse) operation.
- Write the steps in order. List what was done to the secret number.1. divide by 8
2. add 15
The answer was 40. - Work backwards with the opposite operations. Start at the end and undo each step in reverse order.take away 15: 40 − 15 = 25
multiply by 8: 25 × 8 = 200 - Write the number. The number was 200.
Check. 200 ÷ 8 = 25
25 + 15 = 40
We end at 40. ✓
Watch out. Undo the LAST step first.