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Column addition and subtraction puzzles
Multi-digit calculations, inverse operations and missing-digit grids.
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The written summary and downloads below are complete and ready to use.
Column methods let you add and take away big numbers without a calculator. You work on one column at a time, so even a 5-digit sum is just a few small ones.
In this lesson you will learn:
- how to line up numbers in their columns
- column addition, and how to carry
- column subtraction, and how to exchange, even across a zero
- how to use the inverse to check an answer and find a missing number
- how to crack missing-digit puzzles
1. Line up the columns
Key idea
Write the numbers one above the other with ones under ones, tens under tens, and so on.
Line them up from the right.
- A number with fewer digits sits further to the right.
- Always start working from the ones column, on the right.
- On squared paper, put one digit in each square. It keeps the columns straight.
Example: set out 3,456 + 72
ThHTO3456+72
The 2 goes under the 6 (both ones). The 7 goes under the 5 (both tens).
Watch out
If you line up from the left, the 7 lands in the thousands column and 72 turns into 7,200 by mistake.
Line them up: slide the bottom number into the right columns
Things to try
- Drag the bottom number, or use the arrow buttons.
- Watch what the bottom number is really worth as it moves.
2. Column addition and carrying
Key idea
Add each column, starting with the ones. If a column makes 10 or more, write the
ones digit and carry the ten into the next column.
Example: 4,658 + 2,875
ThHTO4658+28757533111
- Ones: 8 + 5 = 13. Write 3, carry 1 into the tens.
- Tens: 5 + 7 + 1 = 13. Write 3, carry 1 into the hundreds.
- Hundreds: 6 + 8 + 1 = 15. Write 5, carry 1 into the thousands.
- Thousands: 4 + 2 + 1 = 7. Write 7.
Answer7,533
Why do we carry?
13 ones is the same as 1 ten and 3 ones. The 3 stays in the ones column. The ten moves to the tens
column, where tens belong. We write the carried 1 small, under the line, so we remember to add it.
- If the last column makes 10 or more, the carried 1 drops straight into the answer: 7,000 + 5,000 = 12,000.
- Adding three numbers? A column can make 20 or more. Then you carry 2.
The column workshop: addition
How it works
- Tap the digit that goes in the glowing box. Work from the ones, on the right.
- Get two sums right with no slips and the workshop moves you up a level. Level 5 adds three numbers.
3. Column subtraction and exchanging
Key idea
Take the bottom digit away from the top digit, starting with the ones. If the top digit is too small,
exchange: take 1 from the next column and make this column 10 bigger.
Example: 6,352 − 2,817
ThHTO5134126352−28173535
- Ones: 2 − 7? 2 is too small. Exchange 1 ten: the 5 tens become 4, the 2 ones become 12. 12 − 7 = 5.
- Tens: 4 − 1 = 3.
- Hundreds: 3 − 8? 3 is too small. Exchange 1 thousand: the 6 becomes 5, the 3 becomes 13. 13 − 8 = 5.
- Thousands: 5 − 2 = 3.
Answer3,535
Watch out
Never take the top digit from the bottom one. 2 − 7 is not 7 − 2. Exchange instead.
Exchanging across a zero: 5,004 − 1,328
ThHTO499145004−13283676
- Ones: 4 − 8? Too small, and the tens and hundreds have nothing to give.
- Go to the thousands. 5 thousands become 4, so the hundreds get 10.
- The hundreds give 1 to the tens: 10 becomes 9, and the tens get 10.
- The tens give 1 to the ones: 10 becomes 9, and the ones become 14.
- Now subtract: 14 − 8 = 6, 9 − 2 = 7, 9 − 3 = 6, 4 − 1 = 3.
Answer3,676
A way to remember
When you exchange across zeros, every zero you pass through becomes a 9.
The column workshop: subtraction
How it works
- If the top digit is big enough, tap the answer digit. If it is too small, tap Exchange first.
- Levels 4 and 5 make you exchange across zeros.
4. Inverse operations: checking and missing numbers
Key idea
Addition and subtraction are inverse operations: one undoes the other.
Three numbers that add up make a fact family of four facts:
345 + 178 = 523178 + 345 = 523
523 − 178 = 345523 − 345 = 178
- Check a subtraction by adding: 523 − 178 = 345, so 345 + 178 should make 523. It does.
- Check an addition by taking away: 345 + 178 = 523, so 523 − 178 should leave 345.
The inverse also finds a missing number:
| Missing number | Do this | So ? is |
|---|---|---|
| ? + 268 = 715 | 715 − 268 | 447 |
| ? − 145 = 380 | 380 + 145 | 525 |
| 600 − ? = 235 | 600 − 235 | 365 |
Watch out
600 − ? = 235 is the trap. The ? is the amount taken away, so it is 600 − 235, not
600 + 235.
Find the missing number
5. Missing-digit puzzles
Key idea
Solve one column at a time, starting with the ones. In each column ask: which digit
makes this column work? Remember any carry.
Example: find the missing digits
HTO4?7+25?713
- Ones: 7 + ? ends in 3. 7 + 6 = 13, so the ? is 6, and 1 is carried.
- Tens: ? + 5 + 1 (carried) ends in 1. 5 + 5 + 1 = 11, so the ? is 5, and 1 is carried.
- Hundreds: 4 + 2 + 1 = 7. That matches the 7 in the answer.
Answer457 + 256 = 713
Subtraction puzzle?
Turn it into an addition with the inverse: answer + bottom number = top number. Then solve it the same way.
- Check your finished puzzle by doing the whole sum. Every column must work.
Crack the grid: fill in the missing digits
How it works
- Tap a ? box, then tap a digit. The ones column is picked for you first.
- Crack two grids with no slips to go up a level. Higher levels have more missing digits.
6. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Use rough paper and set each one out in columns. Move up when you are getting most of them right.
The mistakes that cost marks
- Lining up from the left. 3,456 + 72 set out with the 7 under the 3 gives 10,656. line up from the right: ones under ones. 3,456 + 72 = 3,528.
- Forgetting to carry. 58 + 27 written as 75: the 5 ones and the extra ten have gone missing. 8 + 7 = 15. Write 5, carry 1. 5 + 2 + 1 = 8. So 58 + 27 = 85.
- Taking the smaller digit from the bigger one. 52 − 27 written as 35, because 7 − 2 = 5. 2 − 7 cannot be done, so exchange: 12 − 7 = 5, then 4 − 2 = 2. So 52 − 27 = 25.
- Forgetting the column you exchanged from. 63 − 28 written as 45: the ones became 13, but the 6 tens were left as 6. cross out the 6 and write 5 above it straight away. 13 − 8 = 5, 5 − 2 = 3, so 63 − 28 = 35.
- Turning a zero into 10 instead of 9. In 5,004 − 1,328 the zeros you pass through give 1 away. every zero you exchange across becomes 9.
- Using the wrong inverse. 600 − ? = 235 is not 600 + 235. the ? is what was taken away: 600 − 235 = 365. Check: 600 − 365 = 235.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 2.1.
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