Year 5 Column Addition and Subtraction: check-up and SATs-style questions

Key words. Column addition is a written method. You write the numbers one under the other, with ones under ones, tens under tens and so on, and add one column at a time. When a column adds up to 10 or more, you carry to the next column. The sum or total is the answer to an addition.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

1. Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.

   	TTh	Th	H	T	O
   	4	8	3	6	5
   +	2	7	4	9	8

2. Add the ones. Always start with the ones, on the right.
   5 + 8 = 13
   13 ones is 1 ten and 3 ones. Write 3 in the ones column and carry 1 to the tens column.
3. Add the tens. Now the next column to the left. Add the 1 we carried as well.
   6 + 9 + 1 = 16  (the red digit is the carry)
   16 tens is 1 hundred and 6 tens. Write 6 in the tens column and carry 1 to the hundreds column.
4. Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
   3 + 4 + 1 = 8  (the red digit is the carry)
   8 is less than 10, so write 8 in the hundreds column. There is nothing to carry.
5. Add the thousands. Now the next column to the left.
   8 + 7 = 15
   15 thousands is 1 ten thousand and 5 thousands. Write 5 in the thousands column and carry 1 to the ten thousands column.
6. Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
   4 + 2 + 1 = 7  (the red digit is the carry)
   7 is less than 10, so write 7 in the ten thousands column. There is nothing to carry.
7. Read the answer. Check that the answer has a sensible number of digits, then write it.

   	TTh	Th	H	T	O
   	4	8	3	6	5
   +	2	7	4	9	8
   	7	5	8	6	3
   	1		1	1

   
   48,365 + 27,498 = 75,863

Check. Use the opposite (inverse) operation: 75,863 − 27,498 = 48,365. ✓
Estimate: 48,000 + 27,000 = 75,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Key words. Column subtraction is a written method. You write the bigger number on top and take away one column at a time. If the top digit is too small, you exchange: swap 1 of the next column for 10 of this column. The difference is the answer to a subtraction.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

1. Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.

   	TTh	Th	H	T	O
   	7	3	0	0	4
   −	2	8	6	7	5

2. Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
   4 ones is not enough to take away 5. We need to exchange.
   The tens and hundreds have 0, so there is nothing to exchange from there. We start further to the left, at the thousands.
   Exchange 1 thousand for 10 hundreds: 3 thousands become 2 thousands, and 0 hundreds become 10 hundreds.
   Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 0 tens become 10 tens.
   Exchange 1 ten for 10 ones: 10 tens become 9 tens, and 4 ones become 14 ones.
   14 − 5 = 9
3. Subtract the tens. Now the next column to the left.
   (This digit is 9 now, not 0, because we exchanged from this column.)
   9 − 7 = 2
4. Subtract the hundreds. Now the next column to the left.
   (This digit is 9 now, not 0, because we exchanged from this column.)
   9 − 6 = 3
5. Exchange, then subtract the thousands. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
   2 thousands is not enough to take away 8. We need to exchange.
   (There were 3 thousands to start with, but we gave one away in an earlier exchange.)
   Exchange 1 ten thousand for 10 thousands: 7 ten thousands become 6 ten thousands, and 2 thousands become 12 thousands.
   12 − 8 = 4
6. Subtract the ten thousands. Now the next column to the left.
   (This digit is 6 now, not 7, because we exchanged from this column.)
   6 − 2 = 4
7. Read the answer. Check that the answer has a sensible number of digits, then write it.

   	TTh	Th	H	T	O
   	67	123	90	90	144
   −	2	8	6	7	5
   	4	4	3	2	9

   
   73,004 − 28,675 = 44,329

Check. Use the opposite (inverse) operation: 44,329 + 28,675 = 73,004. ✓
Estimate: 73,000 − 29,000 = 44,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Key words. Column addition is a written method. You write the numbers one under the other, with ones under ones, tens under tens and so on, and add one column at a time. When a column adds up to 10 or more, you carry to the next column. The sum or total is the answer to an addition.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

1. Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value. The numbers have different lengths, so line them up on the right.

   	HTh	TTh	Th	H	T	O
   		1	2	4	5	0
   +			8	7	9	5
   +	1	3	6	2	0	5

2. Add the ones. Always start with the ones, on the right.
   0 + 5 + 5 = 10
   10 ones is 1 ten and 0 ones. Write 0 in the ones column and carry 1 to the tens column.
3. Add the tens. Now the next column to the left. Add the 1 we carried as well.
   5 + 9 + 0 + 1 = 15  (the red digit is the carry)
   15 tens is 1 hundred and 5 tens. Write 5 in the tens column and carry 1 to the hundreds column.
4. Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
   4 + 7 + 2 + 1 = 14  (the red digit is the carry)
   14 hundreds is 1 thousand and 4 hundreds. Write 4 in the hundreds column and carry 1 to the thousands column.
5. Add the thousands. Now the next column to the left. Add the 1 we carried as well.
   2 + 8 + 6 + 1 = 17  (the red digit is the carry)
   17 thousands is 1 ten thousand and 7 thousands. Write 7 in the thousands column and carry 1 to the ten thousands column.
6. Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
   1 + 3 + 1 = 5  (the red digit is the carry)
   5 is less than 10, so write 5 in the ten thousands column. There is nothing to carry.
7. Bring down the hundred thousands digit. The other numbers have no digit in this column, so there is nothing to add.
   There is only one digit here, so write 1 in the hundred thousands column.
8. Read the answer. Check that the answer has a sensible number of digits, then write it.

   	HTh	TTh	Th	H	T	O
   		1	2	4	5	0
   +			8	7	9	5
   +	1	3	6	2	0	5
   	1	5	7	4	5	0
   		1	1	1	1

   
   12,450 + 8,795 + 136,205 = 157,450

Check. Use the opposite (inverse) operation: 157,450 − 8,795 − 136,205 = 12,450. ✓
Estimate: 12,500 + 8,800 + 136,200 = 157,500 (rounding to the nearest 100). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Key words. To compensate is to make up for a change. You use an easier number, then correct the answer.

Method. Compensate: take away the nearby round number 2,000 instead, then fix the answer.

1. Spot a nearby round number. 2,000 is much easier to work with than 1,999.
   1,999 is 1 less than 2,000:  1,999 = 2,000 − 1
2. Take away 2,000 instead of 1,999. Round numbers are easy.
   65,300 − 2,000 = 63,300
3. Fix the answer. We took away 1 too many, so we must put 1 back.
   63,300 + 1 = 63,301

Check. Use the opposite operation: 63,301 + 1,999 = 65,300. ✓

Watch out. Do not forget the last step. You changed the number you add or take away, so the answer needs fixing.

Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.

Method. Round each number to the nearest 10,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.

1. Round 52,860 to the nearest 10,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.
   52,860 ≈ 50,000  (the thousands digit is 2, so round down)
2. Round 38,140 to the nearest 10,000. Do the same for the next number.
   38,140 ≈ 40,000  (the thousands digit is 8, so round up)
3. Do the easy calculation. Rounded numbers are easy to add or subtract.
   50,000 + 40,000 = 90,000

Check. The exact answer is 91,000. It is close to the estimate of 90,000. ✓

Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).

Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.

1. Look at the number sentence. The box stands for the number we want to find.
   500,000 − ? = 273,450
2. Draw a bar model. The whole is the first number. The missing number is one of the parts.
3. Choose the operation. To find a missing part, take the part we know away from the whole.
   500,000 − 273,450
4. Work it out. Use the column method if the numbers are big.
   500,000 − 273,450 = 226,550
   

   	HTh	TTh	Th	H	T	O
   	45	90	90	90	100	0
   −	2	7	3	4	5	0
   	2	2	6	5	5	0

5. Check in the number sentence. Put the answer back in the box.
   500,000 − 226,550 = 273,450 ✓

Check. Read the whole number sentence with your answer in the box. It must be true. ✓

Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.

Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.

Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.

1. Write Tom’s answer as an inverse calculation. The opposite operation should take us back to the first number.
   62,845 + 29,576 = 92,321  so  92,321 − 29,576 should be 62,845
2. Work it out. Use the column method.
   92,321 − 29,576 = 62,745
   

   	TTh	Th	H	T	O
   	89	112	123	112	111
   −	2	9	5	7	6
   	6	2	7	4	5

3. Compare. We should get 62,845.
   We got 62,745. 62,745 is not 62,845, so Tom is wrong.
4. Find the correct answer. Do the original calculation again.
   62,845 + 29,576 = 92,421
   

   	TTh	Th	H	T	O
   	6	2	8	4	5
   +	2	9	5	7	6
   	9	2	4	2	1
   	1	1	1	1

Check. The correct answer is 92,421. 92,421 − 29,576 = 62,845. ✓

Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.

Key words. A comparison bar model shows two amounts side by side. The difference is how much longer one bar is than the other.

Method. Draw a comparison bar model. The longer bar is the bigger amount. The dashed part is the difference. Then decide whether to add or subtract.

1. Draw a comparison bar model. The two bars let us see which amount is bigger and what the difference is.
2. Choose the operation. We want the difference between the two amounts, so we take the smaller from the bigger.
   125,400 − 87,650
3. Work it out. Use the column method because the numbers are big.
   125,400 − 87,650 = 37,750
   

   	HTh	TTh	Th	H	T	O
   	01	112	145	134	100	0
   −		8	7	6	5	0
   		3	7	7	5	0

4. Write the answer.
   37,750 people

Check. Use the opposite operation: 87,650 + 37,750 = 125,400. ✓

Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.

Method. This is a two-step problem. First add up everything that was used. Then take that total away from the amount at the start.

1. Draw a bar model. The whole bar is 200,000. It is split into the two amounts sold and the part that is left.
2. Step 1: add the two amounts. Put the two parts together.
   65,400 + 48,750 = 114,150
   

   	HTh	TTh	Th	H	T	O
   		6	5	4	0	0
   +		4	8	7	5	0
   	1	1	4	1	5	0
   	1	1	1

3. Step 2: take the total away from the start. What is left is the whole minus the parts that were used.
   200,000 − 114,150 = 85,850
   

   	HTh	TTh	Th	H	T	O
   	12	90	90	90	100	0
   −	1	1	4	1	5	0
   		8	5	8	5	0

4. Write the answer.
   85,850 tickets

Check. Add everything back: 65,400 + 48,750 + 85,850 = 200,000, which is the start amount. ✓

Watch out. Do not take away just one of the amounts. Both amounts were used, so take away the total of both.

Key words. A digit is one of 0 to 9. The ones column is the column on the right.

Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.

1. Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.
   62,450 − 28,716 = 33,734  means  28,716 + 33,734 = 62,450
   

   	TTh	Th	H	T	O
   	2	?	7	1	6
   +	3	3	7	3	?
   	6	2	?	5	0

2. Find the missing digit in the ones column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?
   ? + 6 must end in 0.
   Count on from 6 to a number that ends in 0: 6 + 4 = 10.
   The missing digit is 4. Carry 1 to the tens column.
3. Look at the tens column. Nothing is hidden here.
   1 + 3 + 1 = 5. Write 5.
4. Find the missing digit in the hundreds column of the top number. We know the digits we are adding, so we add them (and the carry).
   7 + 7 = 14. The missing digit is 4. Carry 1 to the thousands column.
5. Find the missing digit in the thousands column of the bottom number. The answer digit is 2, so the column total must end in 2. What do we add to the digits we know?
   ? + 3 + 1 must end in 2.
   Count on from 4 to a number that ends in 2: 4 + 8 = 12.
   The missing digit is 8. Carry 1 to the ten thousands column.
6. Look at the ten thousands column. Nothing is hidden here.
   2 + 3 + 1 = 6. Write 6.
7. Write the completed calculation. Put the missing digits back and read the whole calculation.

   	TTh	Th	H	T	O
   	56	112	144	45	100
   −	2	8	7	1	6
   	3	3	7	3	4

   
   62,450 − 28,716 = 33,734
   answer, ones digit: 4
   top number, hundreds digit: 4
   bottom number, thousands digit: 8

Check. Subtract again with the completed numbers: 62,450 − 28,716 = 33,734. ✓

Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.

Key words. Column addition is a written method. You write the numbers one under the other, with ones under ones, tens under tens and so on, and add one column at a time. When a column adds up to 10 or more, you carry to the next column. The sum or total is the answer to an addition.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

1. Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.

   	HTh	TTh	Th	H	T	O
   	4	5	6	2	0	8
   +	2	3	5	7	9	4

2. Add the ones. Always start with the ones, on the right.
   8 + 4 = 12
   12 ones is 1 ten and 2 ones. Write 2 in the ones column and carry 1 to the tens column.
3. Add the tens. Now the next column to the left. Add the 1 we carried as well.
   0 + 9 + 1 = 10  (the red digit is the carry)
   10 tens is 1 hundred and 0 tens. Write 0 in the tens column and carry 1 to the hundreds column.
4. Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
   2 + 7 + 1 = 10  (the red digit is the carry)
   10 hundreds is 1 thousand and 0 hundreds. Write 0 in the hundreds column and carry 1 to the thousands column.
5. Add the thousands. Now the next column to the left. Add the 1 we carried as well.
   6 + 5 + 1 = 12  (the red digit is the carry)
   12 thousands is 1 ten thousand and 2 thousands. Write 2 in the thousands column and carry 1 to the ten thousands column.
6. Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
   5 + 3 + 1 = 9  (the red digit is the carry)
   9 is less than 10, so write 9 in the ten thousands column. There is nothing to carry.
7. Add the hundred thousands. Now the next column to the left.
   4 + 2 = 6
   6 is less than 10, so write 6 in the hundred thousands column. There is nothing to carry.
8. Read the answer. Check that the answer has a sensible number of digits, then write it.

   	HTh	TTh	Th	H	T	O
   	4	5	6	2	0	8
   +	2	3	5	7	9	4
   	6	9	2	0	0	2
   		1	1	1	1

   
   456,208 + 235,794 = 692,002

Check. Use the opposite (inverse) operation: 692,002 − 235,794 = 456,208. ✓
Estimate: 460,000 + 240,000 = 700,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Key words. Column subtraction is a written method. You write the bigger number on top and take away one column at a time. If the top digit is too small, you exchange: swap 1 of the next column for 10 of this column. The difference is the answer to a subtraction.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

1. Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.

   	HTh	TTh	Th	H	T	O
   	8	0	0	0	0	0
   −	3	5	4	2	7	8

2. Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
   0 ones is not enough to take away 8. We need to exchange.
   The tens, hundreds, thousands and ten thousands have 0, so there is nothing to exchange from there. We start further to the left, at the hundred thousands.
   Exchange 1 hundred thousand for 10 ten thousands: 8 hundred thousands become 7 hundred thousands, and 0 ten thousands become 10 ten thousands.
   Exchange 1 ten thousand for 10 thousands: 10 ten thousands become 9 ten thousands, and 0 thousands become 10 thousands.
   Exchange 1 thousand for 10 hundreds: 10 thousands become 9 thousands, and 0 hundreds become 10 hundreds.
   Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 0 tens become 10 tens.
   Exchange 1 ten for 10 ones: 10 tens become 9 tens, and 0 ones become 10 ones.
   10 − 8 = 2
3. Subtract the tens. Now the next column to the left.
   (This digit is 9 now, not 0, because we exchanged from this column.)
   9 − 7 = 2
4. Subtract the hundreds. Now the next column to the left.
   (This digit is 9 now, not 0, because we exchanged from this column.)
   9 − 2 = 7
5. Subtract the thousands. Now the next column to the left.
   (This digit is 9 now, not 0, because we exchanged from this column.)
   9 − 4 = 5
6. Subtract the ten thousands. Now the next column to the left.
   (This digit is 9 now, not 0, because we exchanged from this column.)
   9 − 5 = 4
7. Subtract the hundred thousands. Now the next column to the left.
   (This digit is 7 now, not 8, because we exchanged from this column.)
   7 − 3 = 4
8. Read the answer. Check that the answer has a sensible number of digits, then write it.

   	HTh	TTh	Th	H	T	O
   	78	90	90	90	90	100
   −	3	5	4	2	7	8
   	4	4	5	7	2	2

   
   800,000 − 354,278 = 445,722

Check. Use the opposite (inverse) operation: 445,722 + 354,278 = 800,000. ✓
Estimate: 800,000 − 350,000 = 450,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Method. Round the numbers, work out an easy estimate, then find the choice that is closest.

1. Round the numbers to the nearest 1,000. Rounded numbers are quick to add or subtract.
   48,920 ≈ 49,000  (the hundreds digit is 9, so round up)
   31,070 ≈ 31,000  (the hundreds digit is 0, so round down)
2. Work out the estimate.
   49,000 + 31,000 = 80,000
3. Choose the closest answer. The choices are very different sizes, so only one can be close.
   A: 8,000
   B: 80,000  (close to the estimate)
   C: 800,000
   D: 18,000

Check. The exact answer is 79,990, which is close to 80,000. ✓

Watch out. You do not need the exact answer. Round, work out the easy calculation, and look for the closest choice.

Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.

1. Look at the number sentence. The box stands for the number we want to find.
   ? − 24,500 = 61,250
2. Draw a bar model. The missing number is the whole. It is made of the two parts we know.
3. Choose the operation. To find the whole, add the two parts together.
   61,250 + 24,500
4. Work it out. Use the column method if the numbers are big.
   61,250 + 24,500 = 85,750
   

   	TTh	Th	H	T	O
   	6	1	2	5	0
   +	2	4	5	0	0
   	8	5	7	5	0

5. Check in the number sentence. Put the answer back in the box.
   85,750 − 24,500 = 61,250 ✓

Check. Read the whole number sentence with your answer in the box. It must be true. ✓

Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.

Method. This is a two-step problem. First add up everything that was used. Then take that total away from the amount at the start.

1. Draw a bar model. The whole bar is 65,000. It is split into the two amounts collected and the part that is left.
2. Step 1: add the two amounts. Put the two parts together.
   28,450 + 19,375 = 47,825
   

   	TTh	Th	H	T	O
   	2	8	4	5	0
   +	1	9	3	7	5
   	4	7	8	2	5
   	1		1

3. Step 2: take the total away from the start. What is left is the whole minus the parts that were used.
   65,000 − 47,825 = 17,175
   

   	TTh	Th	H	T	O
   	56	145	90	90	100
   −	4	7	8	2	5
   	1	7	1	7	5

4. Write the answer.
   17,175 bottle tops

Check. Add everything back: 28,450 + 19,375 + 17,175 = 65,000, which is the start amount. ✓

Watch out. Do not take away just one of the amounts. Both amounts were used, so take away the total of both.

Method. Estimate the answer by rounding. If Amira’s answer is nowhere near the estimate, it cannot be right.

1. Round the numbers. Round to the nearest 1,000 so the calculation is easy.
   48,360 ≈ 48,000  (the hundreds digit is 3, so round down)
   29,875 ≈ 30,000  (the hundreds digit is 8, so round up)
2. Work out the estimate.
   48,000 − 30,000 = 18,000
3. Compare with Amira’s answer. Amira got 28,485. The estimate says the answer should be close to 18,000.
   28,485 is a long way from 18,000, so it is not sensible.
4. Find the correct answer. Use the column method carefully.

   	TTh	Th	H	T	O
   	34	178	123	156	100
   −	2	9	8	7	5
   	1	8	4	8	5

   
   48,360 − 29,875 = 18,485

Check. The correct answer 18,485 is close to the estimate 18,000. ✓

Watch out. Always ask: is my answer about the right size? An estimate takes a few seconds and can catch big mistakes.

Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.

Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.

1. Write Sam’s answer as an inverse calculation. The opposite operation should take us back to the first number.
   100,000 − 37,625 = 62,475  so  62,475 + 37,625 should be 100,000
2. Work it out. Use the column method.
   62,475 + 37,625 = 100,100
   

   	HTh	TTh	Th	H	T	O
   		6	2	4	7	5
   +		3	7	6	2	5
   	1	0	0	1	0	0
   	1	1	1	1	1

3. Compare. We should get 100,000.
   We got 100,100. 100,100 is not 100,000, so Sam is wrong.
4. Find the correct answer. Do the original calculation again.
   100,000 − 37,625 = 62,375
   

   	HTh	TTh	Th	H	T	O
   	01	90	90	90	90	100
   −		3	7	6	2	5
   		6	2	3	7	5

Check. The correct answer is 62,375. 62,375 + 37,625 = 100,000. ✓

Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.

Key words. To work backwards is to start from the answer and use the inverse of each step in reverse order.

Method. Work backwards. Use the inverse (opposite) of each step, starting from the last one.

1. Write the steps in order. Dev did: start → subtract 12,500 → add 30,000 → 100,000.
   ? → − 12,500 → + 30,000 → 100,000
2. Undo the last step. The last step was + 30,000. The inverse is − 30,000.
   100,000 − 30,000 = 70,000
   

   	HTh	TTh	Th	H	T	O
   	01	100	0	0	0	0
   −		3	0	0	0	0
   		7	0	0	0	0

3. Undo the first step. The first step was − 12,500. The inverse is + 12,500.
   70,000 + 12,500 = 82,500
   

   	TTh	Th	H	T	O
   	7	0	0	0	0
   +	1	2	5	0	0
   	8	2	5	0	0

4. Write the number.
   82,500

Check. Go forward: 82,500 − 12,500 = 70,000 and 70,000 + 30,000 = 100,000. ✓

Watch out. 3 marks: 1 for undoing the addition, 1 for undoing the subtraction and 1 for the correct answer. Work backwards, not forwards.

Method. Find how many points Priya has. Add Noah and Priya. Then take that away from the total to find Leo.

1. Draw a bar model. The whole bar is 150,000. It has three parts: Noah, Priya and Leo.
2. Step 1: find Priya’s points. Priya has 12,500 more than Noah, so add.
   38,250 + 12,500 = 50,750
   

   	TTh	Th	H	T	O
   	3	8	2	5	0
   +	1	2	5	0	0
   	5	0	7	5	0
   	1

3. Step 2: add Noah and Priya. We need to know how many points they have together.
   38,250 + 50,750 = 89,000
   

   	TTh	Th	H	T	O
   	3	8	2	5	0
   +	5	0	7	5	0
   	8	9	0	0	0
   		1	1

4. Step 3: take them from the total. Leo has the rest.
   150,000 − 89,000 = 61,000
   

   	HTh	TTh	Th	H	T	O
   	01	145	100	0	0	0
   −		8	9	0	0	0
   		6	1	0	0	0

5. Write the answer.
   61,000 points

Check. 38,250 + 50,750 + 61,000 = 150,000. ✓

Watch out. 3 marks: 1 for finding Priya’s points, 1 for adding Noah and Priya and 1 for the correct answer.