Year 6: Mental calculation with mixed operations and large numbers
Part of the Year 6 long multiplication, long division and bodmas chapter. Chapter overview · whole chapter on one page
Mental calculation with mixed operations and large numbers
You do not always need a written method. Good mental methods use facts you already know. Here are four strategies:
- Partition: split one number into easy parts, multiply each part, then add the answers. For example, 8 × 125 = 8 × 100 + 8 × 25.
- Use a near round number: 18 × 49 is close to 18 × 50. Multiply by 50, then take away one lot of 18.
- Halve and double: 16 × 35 = 8 × 70, because halving one number and doubling the other keeps the answer the same.
- Use zeros: 4,000 × 60 = 4 × 6 with 4 zeros. Take the zeros off first.
In a calculation with mixed operations, work out each part first, then add and subtract. Remember the order of operations (section 2.5).
Key ideas
- Partition one number into easy parts, then multiply each part.
- Use a nearby round number, then correct for the small difference.
- Halving one number and doubling the other does not change the product.
- Take zeros off, multiply, then put them back.
- In mixed calculations, multiply and divide before you add and subtract.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Split one number into parts, multiply each part, then add the answers.
- Split 125 into parts. Parts that are easy to multiply by 8 make the working simpler.125 = 100 + 25
- Multiply each part by 8. Multiply each part on its own.8 × 100 = 800
8 × 25 = 200 - Add the answers. Add the partial answers together.800 + 200 = 1,000
Check. Estimate: 8 × 100 = 800, which is close to 1,000. ✓
Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Multiply by a round number that is close, then correct the answer for the small difference.
- Use the nearby round number, 50. 49 is 1 less than 50, and 50 is easy to multiply by.49 = 50 − 1
- Multiply 18 by 50. Multiply by the round number first.18 × 50 = 900
- Correct for the difference. 49 is 1 less than 50, so we take away 1 lots of 18.18 × 1 = 18, so 900 − 18 = 882
Check. The estimate 20 × 50 = 1,000 is close to 882. ✓
Watch out. Correct in the right direction. If the round number was too big, take the difference away.
Method. Split the number into easy parts, divide each part, then add the answers.
- Split the number into parts. Choose parts of 1,440 that 24 divides into exactly.1,440 = 1,200 + 240
- Divide each part by 24. Each answer is a whole number, so the parts are easy to work out.1,200 ÷ 24 = 50
240 ÷ 24 = 10 - Add the answers. Add the answers from each part.50 + 10 = 60
Check. 24 × 60 = 1,440. ✓
Watch out. Pick parts that the divisor fits into exactly. Then each part is a simple division.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Halve one number and double the other. The product stays the same.
- Halve one number and double the other. Halving 16 and doubling 35 does not change the answer.16 × 35 = 8 × 70
- Work out 8 × 70. If 70 is hard to multiply, split it into easy parts and add the answers.8 × 70 = 560
- Check the answer. Check by halving and doubling in the other way, or by estimating.Estimate: 20 × 40 = 800, which is close to 560
Check. Check with the other order: 35 × 16 = 560. ✓
Watch out. Only halve and double in the same way. If you halve 16 you must double 35.
Watch out
When you round a number, remember to correct the answer. If you used 50 instead of 49, you used one lot too many, so take one lot away. Also, do not put the zeros in the wrong place: 4,000 × 60 is 240,000, not 24,000.Exercise 2D Mental calculation and mixed operations
Non-calculatorKey words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Split one number into parts, multiply each part, then add the answers.
- Split 36 into parts. Parts that are easy to multiply by 5 make the working simpler.36 = 30 + 6
- Multiply each part by 5. Multiply each part on its own.5 × 30 = 150
5 × 6 = 30 - Add the answers. Add the partial answers together.150 + 30 = 180
Check. Estimate: 5 × 40 = 200, which is close to 180. ✓
Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Split one number into parts, multiply each part, then add the answers.
- Split 48 into parts. Parts that are easy to multiply by 4 make the working simpler.48 = 40 + 8
- Multiply each part by 4. Multiply each part on its own.4 × 40 = 160
4 × 8 = 32 - Add the answers. Add the partial answers together.160 + 32 = 192
Check. Estimate: 4 × 50 = 200, which is close to 192. ✓
Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Split one number into parts, multiply each part, then add the answers.
- Split 73 into parts. Parts that are easy to multiply by 6 make the working simpler.73 = 70 + 3
- Multiply each part by 6. Multiply each part on its own.6 × 70 = 420
6 × 3 = 18 - Add the answers. Add the partial answers together.420 + 18 = 438
Check. Estimate: 6 × 70 = 420, which is close to 438. ✓
Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Multiply by a round number that is close, then correct the answer for the small difference.
- Use the nearby round number, 100. 99 is 1 less than 100, and 100 is easy to multiply by.99 = 100 − 1
- Multiply 7 by 100. Multiply by the round number first.7 × 100 = 700
- Correct for the difference. 99 is 1 less than 100, so we take away 1 lots of 7.7 × 1 = 7, so 700 − 7 = 693
Check. The estimate 7 × 100 = 700 is close to 693. ✓
Watch out. Correct in the right direction. If the round number was too big, take the difference away.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Multiply by a round number that is close, then correct the answer for the small difference.
- Use the nearby round number, 50. 46 is 4 less than 50, and 50 is easy to multiply by.46 = 50 − 4
- Multiply 9 by 50. Multiply by the round number first.9 × 50 = 450
- Correct for the difference. 46 is 4 less than 50, so we take away 4 lots of 9.9 × 4 = 36, so 450 − 36 = 414
Check. The estimate 9 × 50 = 450 is close to 414. ✓
Watch out. Correct in the right direction. If the round number was too big, take the difference away.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Split one number into parts, multiply each part, then add the answers.
- Split 50 into parts. Parts that are easy to multiply by 14 make the working simpler.50 = 40 + 10
- Multiply each part by 14. Multiply each part on its own.14 × 40 = 560
14 × 10 = 140 - Add the answers. Add the partial answers together.560 + 140 = 700
Check. Estimate: 10 × 50 = 500, which is close to 700. ✓
Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Multiply by a round number that is close, then correct the answer for the small difference.
- Use the nearby round number, 100. 98 is 2 less than 100, and 100 is easy to multiply by.98 = 100 − 2
- Multiply 6 by 100. Multiply by the round number first.6 × 100 = 600
- Correct for the difference. 98 is 2 less than 100, so we take away 2 lots of 6.6 × 2 = 12, so 600 − 12 = 588
Check. The estimate 6 × 100 = 600 is close to 588. ✓
Watch out. Correct in the right direction. If the round number was too big, take the difference away.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Halve one number and double the other. The product stays the same.
- Halve one number and double the other. Halving 24 and doubling 45 does not change the answer.24 × 45 = 12 × 90
- Work out 12 × 90. If 90 is hard to multiply, split it into easy parts and add the answers.12 × 90 = 1,080
- Check the answer. Check by halving and doubling in the other way, or by estimating.Estimate: 20 × 50 = 1,000, which is close to 1,080
Check. Check with the other order: 45 × 24 = 1,080. ✓
Watch out. Only halve and double in the same way. If you halve 24 you must double 45.
Method. Split the number into easy parts, divide each part, then add the answers.
- Split the number into parts. Choose parts of 3,240 that 18 divides into exactly.3,240 = 1,800 + 1,440
- Divide each part by 18. Each answer is a whole number, so the parts are easy to work out.1,800 ÷ 18 = 100
1,440 ÷ 18 = 80 - Add the answers. Add the answers from each part.100 + 80 = 180
Check. 18 × 180 = 3,240. ✓
Watch out. Pick parts that the divisor fits into exactly. Then each part is a simple division.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Take off the zeros, multiply the digits, then put the zeros back at the end.
- Take the zeros off both numbers. Zeros at the end only move the digits, so we can multiply the digits first.4,000 = 4 × 1,000, and 60 = 6 × 10
- Multiply the digits. Use a times table fact you know.4 × 6 = 24
- Put the zeros back. There are 4 zeros in total, so write 4 zeros after the answer.24 with 4 zeros = 240,000
Check. Check by estimating: 4,000 is close to 4,000, so the answer is about 240,000. ✓
Watch out. Count every zero at the end of both numbers. Put them all back at the end of the answer.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Take off the zeros, multiply the digits, then put the zeros back at the end.
- Take the zeros off both numbers. Zeros at the end only move the digits, so we can multiply the digits first.300 = 3 × 100, and 700 = 7 × 100
- Multiply the digits. Use a times table fact you know.3 × 7 = 21
- Put the zeros back. There are 4 zeros in total, so write 4 zeros after the answer.21 with 4 zeros = 210,000
Check. Check by estimating: 300 is close to 300, so the answer is about 210,000. ✓
Watch out. Count every zero at the end of both numbers. Put them all back at the end of the answer.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Take off the zeros, multiply the digits, then put the zeros back at the end.
- Take the zeros off both numbers. Zeros at the end only move the digits, so we can multiply the digits first.90 = 9 × 10, and 2,000 = 2 × 1,000
- Multiply the digits. Use a times table fact you know.9 × 2 = 18
- Put the zeros back. There are 4 zeros in total, so write 4 zeros after the answer.18 with 4 zeros = 180,000
Check. Check by estimating: 90 is close to 90, so the answer is about 180,000. ✓
Watch out. Count every zero at the end of both numbers. Put them all back at the end of the answer.
Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).
Method. Work out each part with multiplication and division first, then add and subtract from left to right.
- Work out 36 × 5. 36 × 5: 30 × 5 = 150 and 6 × 5 = 30, so 150 + 30 = 180.36 × 5 = 180
- Work out 4 × 25. Four 25s make 100: 25 + 25 = 50, and 50 + 50 = 100.4 × 25 = 100
- Add and subtract the answers. Now the multiplication and division are done, so work from the left to add and subtract.180 + 100 = 280
Check. Check: 36 × 5 is about 40 × 5 = 200, so 280 is sensible. ✓
Watch out. Work out each multiplication first, then add.
Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).
Method. Work out each part with multiplication and division first, then add and subtract from left to right.
- Work out 60 ÷ 4. 60 is 4 lots of 15, because 4 × 15 = 60.60 ÷ 4 = 15
- Work out 12 × 3. 12 × 3 = 30 + 6 = 36.12 × 3 = 36
- Add and subtract the answers. Now the multiplication and division are done, so work from the left to add and subtract.15 + 36 = 51
Check. Check: 15 + 36 = 51. ✓
Watch out. Divide and multiply first, then add.
Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).
Method. Work out each part with multiplication and division first, then add and subtract from left to right.
- Work out 3 × 40. 3 × 40 = 3 × 4 tens = 12 tens = 120.3 × 40 = 120
- Work out 5 × 8. 5 × 8 = 40, because 5 eights make 40.5 × 8 = 40
- Work out 12 ÷ 4. 12 ÷ 4 = 3, because 4 × 3 = 12.12 ÷ 4 = 3
- Add and subtract the answers. Now the multiplication and division are done, so work from the left to add and subtract.120 + 40 − 3 = 157
Check. Check: 120 + 40 = 160, and 160 − 3 = 157. ✓
Watch out. Work out the multiplication and division first. Then add and subtract from left to right.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Split one number into parts, multiply each part, then add the answers.
- Split 25 into parts. Parts that are easy to multiply by 36 make the working simpler.25 = 20 + 5
- Multiply each part by 36. Multiply each part on its own.36 × 20 = 720
36 × 5 = 180 - Add the answers. Add the partial answers together.720 + 180 = 900
Check. Estimate: 40 × 30 = 1,200, which is close to 900. ✓
Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Round 99 to 100. Multiply by 100, then take away one lot of 6.
- Look at the number. Priya used 100, but the question says 99, which is one less.99 = 100 − 1
- Work out 6 × 100. Use the round number first.6 × 100 = 600
- Take away one lot of 6. Because 99 is one less than 100, the answer is one lot of 6 less.600 − 6 = 594, so No. 99 × 6 = 594
Check. 99 × 6 = 594. ✓
Watch out. Check whether a nearby number is exactly the number in the question.
Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.
Method. Split one number into parts, multiply each part, then add the answers.
- Split 1,250 into parts. Parts that are easy to multiply by 8 make the working simpler.1,250 = 1,000 + 250
- Multiply each part by 8. Multiply each part on its own.8 × 1,000 = 8,000
8 × 250 = 2,000 - Add the answers. Add the partial answers together.8,000 + 2,000 = 10,000
Check. Estimate: 8 × 1,000 = 8,000, which is close to 10,000. ✓
Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.