Year 5: Counting in steps of 10, 100, 1,000, 10,000 and 100,000
Part of the Year 5 place value to 1,000,000 chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Counting in steps of 10, 100, 1,000, 10,000 and 100,000
The numbers 10, 100, 1,000, 10,000 and 100,000 are called powers of 10. Each one is 10 times bigger than the one before. When you count on or back in a power of 10, only one digit changes: the digit in the column with the same value as the step.
Count on in 10,000s from 253,400. Look at the table. The ten thousands digit goes up by 1 each time: 2, 3, 4, 5. The other digits stay the same.
| Thousands | Ones | |||||
|---|---|---|---|---|---|---|
| HTh | TTh | Th | H | T | O | |
| 253,400 | 2 | 5 | 3 | 4 | 0 | 0 |
| 263,400 | 2 | 6 | 3 | 4 | 0 | 0 |
| 273,400 | 2 | 7 | 3 | 4 | 0 | 0 |
| 283,400 | 2 | 8 | 3 | 4 | 0 | 0 |
Look out for 9s and 0s. When a digit is 9 and you add 1, it becomes 0 and you carry 1 to the next column. So 10,000 more than 395,000 is 405,000. When a digit is 0 and you take 1 away, you swap 1 from the next column for 10. So 1,000 less than 300,000 is 299,000. The swap can go along more than one column.
Missing numbers. To fill the gaps in a sequence, find two numbers you can see. Count the jumps between them, then divide the difference by the number of jumps. That tells you the size of each jump. To find how many jumps there are between two numbers, divide the difference by the size of one jump.
Key words. A sequence is a list of numbers that follows a rule. A power of 10 is 10, 100, 1,000, 10,000, 100,000 and so on. A term is one number in the sequence.
Key ideas
- Counting in a power of 10 changes only one digit: the one in the column with that value.
- A 9 going up carries 1 to the next column. A 0 going down swaps 1 from the next column.
- For 10 more or less, change the tens digit. For 100, the hundreds digit. For 1,000, 10,000 or 100,000, the thousands, ten thousands or hundred thousands digit.
- To find the step in a sequence: difference ÷ number of jumps.
- To find the number of jumps: total distance ÷ size of one jump.
Key words. Counting in steps of 100 means adding 100 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting on in 100s means adding 100 each time. Only the hundreds digit changes (unless we carry or swap).
- Find the size of each jump. We are counting on in steps of 100, which is 1 hundred.Each jump is +100.
- Work out the numbers one jump at a time. Always start from the number you have just found.3,450 + 100 = 3,550 (the hundreds digit goes 4 → 5)
3,550 + 100 = 3,650 (the hundreds digit goes 5 → 6)
3,650 + 100 = 3,750 (the hundreds digit goes 6 → 7)
3,750 + 100 = 3,850 (the hundreds digit goes 7 → 8) - Write the numbers in order. 3,550, 3,650, 3,750, 3,850
Check. Go the other way from the last number: 3,850 − 100 = 3,750. ✓
Watch out. Only the hundreds digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 1,000 means taking away 1,000 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting back in 1,000s means taking away 1,000 each time. Only the thousands digit changes (unless we carry or swap).
- Find the size of each jump. We are counting back in steps of 1,000, which is 1 thousand.Each jump is −1,000.
- Work out the numbers one jump at a time. Always start from the number you have just found.62,500 − 1,000 = 61,500 (the thousands digit goes 2 → 1)
61,500 − 1,000 = 60,500 (the thousands digit goes 1 → 0)
60,500 − 1,000 = 59,500 (0 thousands: swap 1 from the next column for 10 thousands, then take 1 thousand away)
59,500 − 1,000 = 58,500 (the thousands digit goes 9 → 8)
58,500 − 1,000 = 57,500 (the thousands digit goes 8 → 7) - Write the numbers in order. 61,500, 60,500, 59,500, 58,500, 57,500
Check. Go the other way from the last number: 57,500 + 1,000 = 58,500. ✓
Watch out. Only the thousands digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 100 means adding 100 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting on in 100s means adding 100 each time. Only the hundreds digit changes (unless we carry or swap).
- Find the size of each jump. We are counting on in steps of 100, which is 1 hundred.Each jump is +100.
- Work out the numbers one jump at a time. Always start from the number you have just found.98,600 + 100 = 98,700 (the hundreds digit goes 6 → 7)
98,700 + 100 = 98,800 (the hundreds digit goes 7 → 8)
98,800 + 100 = 98,900 (the hundreds digit goes 8 → 9)
98,900 + 100 = 99,000 (9 hundreds + 1 hundred = 10 hundreds: write 0 and carry 1 to the next column) - Write the numbers in order. 98,700, 98,800, 98,900, 99,000
Check. Go the other way from the last number: 99,000 − 100 = 98,900. ✓
Watch out. Only the hundreds digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 10,000 means taking away 10,000 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting back in 10,000s means taking away 10,000 each time. Only the ten thousands digit changes (unless we carry or swap).
- Find the size of each jump. We are counting back in steps of 10,000, which is 1 ten thousand.Each jump is −10,000.
- Work out the numbers one jump at a time. Always start from the number you have just found.205,000 − 10,000 = 195,000 (0 ten thousands: swap 1 from the next column for 10 ten thousands, then take 1 ten thousand away)
195,000 − 10,000 = 185,000 (the ten thousands digit goes 9 → 8)
185,000 − 10,000 = 175,000 (the ten thousands digit goes 8 → 7) - Write the numbers in order. 195,000, 185,000, 175,000
Check. Go the other way from the last number: 175,000 + 10,000 = 185,000. ✓
Watch out. Only the ten thousands digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Method. To find 100,000 more, change the hundred thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the hundred thousands digit. 100,000 is one hundred thousand, so we change the hundred thousands digit.
Thousands Ones HTh TTh Th H T O 4 7 8 3 5 0 - Add 1 to the hundred thousands digit. 100,000 more means 1 more hundred thousand.4 + 1 = 5 hundred thousands
- Write the new number. The digits that did not change stay the same.478,350 + 100,000 = 578,350
Check. Do the opposite: 578,350 − 100,000 = 478,350. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 1,000 less, change the thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the thousands digit. 1,000 is one thousand, so we change the thousands digit.
Thousands Ones HTh TTh Th H T O 3 0 0 0 0 0 - Take 1 from the thousands digit. There are 0 thousands, so we swap 1 from the next column for 10 thousands. If that next digit is also 0 we swap again. Then take 1 away.0 thousands → swap, then 10 − 1 = 9 thousands
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 300,000 3 0 0 0 0 0 299,000 2 9 9 0 0 0
300,000 − 1,000 = 299,000
Check. Do the opposite: 299,000 + 1,000 = 300,000. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.
- Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.From 132,000 to 152,000 is 2 jumps.
The difference is 20,000.
20,000 ÷ 2 = 10,000 - Say which way and how big each jump is. The numbers are getting bigger, so we count on.Each jump is +10,000.
- Fill in the missing numbers. Add (or take away) one jump from a number you know. Where two gaps are side by side, fill one first and use it for the next.Number 2: 132,000 + 10,000 = 142,000
Number 4: 152,000 + 10,000 = 162,000 - Write the missing numbers. Put them in the order they appear.142,000, 162,000
Check. Read the whole sequence again and check every jump is +10,000: 132,000, 142,000, 152,000, 162,000, 172,000. ✓
Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.
Method. Work out how much each term changes by. If the jump is always the same, that is the rule.
- Write the first terms in a place-value table. Look for the one column that changes.
Thousands Ones TTh Th H T O 36,250 3 6 2 5 0 36,350 3 6 3 5 0 36,450 3 6 4 5 0 - Find the jump between two terms next to each other. The column that changes tells us the size of the jump.36,350 − 36,250 = 100 (it goes up)
- Check the jump is the same all the way along. A rule has to work for every pair of terms.36,350 − 36,250 = 100
36,450 − 36,350 = 100
36,550 − 36,450 = 100 - Write the rule and the next two terms. Add 100 each time.Rule: add 100.
Next two terms: 36,650, 36,750
Check. 36,550 + 100 = 36,650 and 36,650 + 100 = 36,750. ✓
Watch out. Check more than one pair of terms. One pair is not enough to be sure of the rule.
Method. Find the total distance, then work out how many jumps of 10,000 fit into it.
- Find how far it is from the start to the end. Subtract the start from the end.200,000 − 120,000 = 80,000
- Divide by the size of each jump. Each jump is 10,000.80,000 ÷ 10,000 = 8
- Write the answer. 8 jumps of 10,000
Check. 8 × 10,000 = 80,000 and 120,000 + 80,000 = 200,000. ✓
Watch out. Count the jumps, not the numbers you say. Counting the numbers would include the start and give one too many.
Watch out
Do not change every digit. 100,000 less than 640,000 is 540,000, not 630,000 or 539,000. Only the hundred thousands digit changes. Also take care with 9s and 0s: 1,000 less than 300,000 is 299,000, not 200,000.Exercise 1C Counting in powers of 10
Non-calculatorKey words. Counting in steps of 10 means adding 10 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting on in 10s means adding 10 each time. Only the tens digit changes (unless we carry or swap).
- Find the size of each jump. We are counting on in steps of 10, which is 1 ten.Each jump is +10.
- Work out the numbers one jump at a time. Always start from the number you have just found.4,570 + 10 = 4,580 (the tens digit goes 7 → 8)
4,580 + 10 = 4,590 (the tens digit goes 8 → 9)
4,590 + 10 = 4,600 (9 tens + 1 ten = 10 tens: write 0 and carry 1 to the next column) - Write the numbers in order. 4,580, 4,590, 4,600
Check. Go the other way from the last number: 4,600 − 10 = 4,590. ✓
Watch out. Only the tens digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 100 means adding 100 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting on in 100s means adding 100 each time. Only the hundreds digit changes (unless we carry or swap).
- Find the size of each jump. We are counting on in steps of 100, which is 1 hundred.Each jump is +100.
- Work out the numbers one jump at a time. Always start from the number you have just found.28,300 + 100 = 28,400 (the hundreds digit goes 3 → 4)
28,400 + 100 = 28,500 (the hundreds digit goes 4 → 5)
28,500 + 100 = 28,600 (the hundreds digit goes 5 → 6) - Write the numbers in order. 28,400, 28,500, 28,600
Check. Go the other way from the last number: 28,600 − 100 = 28,500. ✓
Watch out. Only the hundreds digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 1,000 means adding 1,000 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting on in 1,000s means adding 1,000 each time. Only the thousands digit changes (unless we carry or swap).
- Find the size of each jump. We are counting on in steps of 1,000, which is 1 thousand.Each jump is +1,000.
- Work out the numbers one jump at a time. Always start from the number you have just found.351,200 + 1,000 = 352,200 (the thousands digit goes 1 → 2)
352,200 + 1,000 = 353,200 (the thousands digit goes 2 → 3)
353,200 + 1,000 = 354,200 (the thousands digit goes 3 → 4) - Write the numbers in order. 352,200, 353,200, 354,200
Check. Go the other way from the last number: 354,200 − 1,000 = 353,200. ✓
Watch out. Only the thousands digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 10,000 means adding 10,000 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting on in 10,000s means adding 10,000 each time. Only the ten thousands digit changes (unless we carry or swap).
- Find the size of each jump. We are counting on in steps of 10,000, which is 1 ten thousand.Each jump is +10,000.
- Work out the numbers one jump at a time. Always start from the number you have just found.420,000 + 10,000 = 430,000 (the ten thousands digit goes 2 → 3)
430,000 + 10,000 = 440,000 (the ten thousands digit goes 3 → 4)
440,000 + 10,000 = 450,000 (the ten thousands digit goes 4 → 5) - Write the numbers in order. 430,000, 440,000, 450,000
Check. Go the other way from the last number: 450,000 − 10,000 = 440,000. ✓
Watch out. Only the ten thousands digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 10 means taking away 10 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting back in 10s means taking away 10 each time. Only the tens digit changes (unless we carry or swap).
- Find the size of each jump. We are counting back in steps of 10, which is 1 ten.Each jump is −10.
- Work out the numbers one jump at a time. Always start from the number you have just found.6,080 − 10 = 6,070 (the tens digit goes 8 → 7)
6,070 − 10 = 6,060 (the tens digit goes 7 → 6)
6,060 − 10 = 6,050 (the tens digit goes 6 → 5) - Write the numbers in order. 6,070, 6,060, 6,050
Check. Go the other way from the last number: 6,050 + 10 = 6,060. ✓
Watch out. Only the tens digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 100 means taking away 100 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting back in 100s means taking away 100 each time. Only the hundreds digit changes (unless we carry or swap).
- Find the size of each jump. We are counting back in steps of 100, which is 1 hundred.Each jump is −100.
- Work out the numbers one jump at a time. Always start from the number you have just found.71,450 − 100 = 71,350 (the hundreds digit goes 4 → 3)
71,350 − 100 = 71,250 (the hundreds digit goes 3 → 2)
71,250 − 100 = 71,150 (the hundreds digit goes 2 → 1) - Write the numbers in order. 71,350, 71,250, 71,150
Check. Go the other way from the last number: 71,150 + 100 = 71,250. ✓
Watch out. Only the hundreds digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 1,000 means taking away 1,000 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting back in 1,000s means taking away 1,000 each time. Only the thousands digit changes (unless we carry or swap).
- Find the size of each jump. We are counting back in steps of 1,000, which is 1 thousand.Each jump is −1,000.
- Work out the numbers one jump at a time. Always start from the number you have just found.530,000 − 1,000 = 529,000 (0 thousands: swap 1 from the next column for 10 thousands, then take 1 thousand away)
529,000 − 1,000 = 528,000 (the thousands digit goes 9 → 8)
528,000 − 1,000 = 527,000 (the thousands digit goes 8 → 7) - Write the numbers in order. 529,000, 528,000, 527,000
Check. Go the other way from the last number: 527,000 + 1,000 = 528,000. ✓
Watch out. Only the thousands digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Key words. Counting in steps of 100,000 means taking away 100,000 again and again. 10, 100, 1,000, 10,000 and 100,000 are powers of 10.
Method. Counting back in 100,000s means taking away 100,000 each time. Only the hundred thousands digit changes (unless we carry or swap).
- Find the size of each jump. We are counting back in steps of 100,000, which is 1 hundred thousand.Each jump is −100,000.
- Work out the numbers one jump at a time. Always start from the number you have just found.900,000 − 100,000 = 800,000 (the hundred thousands digit goes 9 → 8)
800,000 − 100,000 = 700,000 (the hundred thousands digit goes 8 → 7)
700,000 − 100,000 = 600,000 (the hundred thousands digit goes 7 → 6) - Write the numbers in order. 800,000, 700,000, 600,000
Check. Go the other way from the last number: 600,000 + 100,000 = 700,000. ✓
Watch out. Only the hundred thousands digit changes, unless it is a 9 going up (carry) or a 0 going down (swap). Check the digit before every jump.
Method. To find 10 more, change the tens digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the tens digit. 10 is one ten, so we change the tens digit.
Thousands Ones Th H T O 7 9 9 5 - Add 1 to the tens digit. There are 9 tens. 1 more ten makes 10 tens, which is 1 of the next column and 0 tens. We carry (and the carry carries on if the next digit is also 9).9 + 1 = 10 → write 0 in the T column and carry 1
- Write the new number. The digits that did not change stay the same.
Thousands Ones Th H T O 7,995 7 9 9 5 8,005 8 0 0 5
7,995 + 10 = 8,005
Check. Do the opposite: 8,005 − 10 = 7,995. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 100 more, change the hundreds digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the hundreds digit. 100 is one hundred, so we change the hundreds digit.
Thousands Ones TTh Th H T O 5 4 9 5 0 - Add 1 to the hundreds digit. There are 9 hundreds. 1 more hundred makes 10 hundreds, which is 1 of the next column and 0 hundreds. We carry (and the carry carries on if the next digit is also 9).9 + 1 = 10 → write 0 in the H column and carry 1
- Write the new number. The digits that did not change stay the same.
Thousands Ones TTh Th H T O 54,950 5 4 9 5 0 55,050 5 5 0 5 0
54,950 + 100 = 55,050
Check. Do the opposite: 55,050 − 100 = 54,950. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 1,000 more, change the thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the thousands digit. 1,000 is one thousand, so we change the thousands digit.
Thousands Ones HTh TTh Th H T O 2 4 9 5 0 0 - Add 1 to the thousands digit. There are 9 thousands. 1 more thousand makes 10 thousands, which is 1 of the next column and 0 thousands. We carry (and the carry carries on if the next digit is also 9).9 + 1 = 10 → write 0 in the Th column and carry 1
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 249,500 2 4 9 5 0 0 250,500 2 5 0 5 0 0
249,500 + 1,000 = 250,500
Check. Do the opposite: 250,500 − 1,000 = 249,500. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 10,000 more, change the ten thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the ten thousands digit. 10,000 is one ten thousand, so we change the ten thousands digit.
Thousands Ones HTh TTh Th H T O 3 9 5 0 0 0 - Add 1 to the ten thousands digit. There are 9 ten thousands. 1 more ten thousand makes 10 ten thousands, which is 1 of the next column and 0 ten thousands. We carry (and the carry carries on if the next digit is also 9).9 + 1 = 10 → write 0 in the TTh column and carry 1
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 395,000 3 9 5 0 0 0 405,000 4 0 5 0 0 0
395,000 + 10,000 = 405,000
Check. Do the opposite: 405,000 − 10,000 = 395,000. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 100,000 more, change the hundred thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the hundred thousands digit. 100,000 is one hundred thousand, so we change the hundred thousands digit.
Thousands Ones HTh TTh Th H T O 2 9 9 9 9 9 - Add 1 to the hundred thousands digit. 100,000 more means 1 more hundred thousand.2 + 1 = 3 hundred thousands
- Write the new number. The digits that did not change stay the same.299,999 + 100,000 = 399,999
Check. Do the opposite: 399,999 − 100,000 = 299,999. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 10,000 more, change the ten thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the ten thousands digit. 10,000 is one ten thousand, so we change the ten thousands digit.
Thousands Ones HTh TTh Th H T O 8 9 9 0 0 0 - Add 1 to the ten thousands digit. There are 9 ten thousands. 1 more ten thousand makes 10 ten thousands, which is 1 of the next column and 0 ten thousands. We carry (and the carry carries on if the next digit is also 9).9 + 1 = 10 → write 0 in the TTh column and carry 1
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 899,000 8 9 9 0 0 0 909,000 9 0 9 0 0 0
899,000 + 10,000 = 909,000
Check. Do the opposite: 909,000 − 10,000 = 899,000. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 10 less, change the tens digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the tens digit. 10 is one ten, so we change the tens digit.
Thousands Ones Th H T O 5 0 0 0 - Take 1 from the tens digit. There are 0 tens, so we swap 1 from the next column for 10 tens. If that next digit is also 0 we swap again. Then take 1 away.0 tens → swap, then 10 − 1 = 9 tens
- Write the new number. The digits that did not change stay the same.
Thousands Ones Th H T O 5,000 5 0 0 0 4,990 4 9 9 0
5,000 − 10 = 4,990
Check. Do the opposite: 4,990 + 10 = 5,000. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 100 less, change the hundreds digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the hundreds digit. 100 is one hundred, so we change the hundreds digit.
Thousands Ones TTh Th H T O 3 0 0 5 0 - Take 1 from the hundreds digit. There are 0 hundreds, so we swap 1 from the next column for 10 hundreds. If that next digit is also 0 we swap again. Then take 1 away.0 hundreds → swap, then 10 − 1 = 9 hundreds
- Write the new number. The digits that did not change stay the same.
Thousands Ones TTh Th H T O 30,050 3 0 0 5 0 29,950 2 9 9 5 0
30,050 − 100 = 29,950
Check. Do the opposite: 29,950 + 100 = 30,050. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 1,000 less, change the thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the thousands digit. 1,000 is one thousand, so we change the thousands digit.
Thousands Ones HTh TTh Th H T O 1 0 0 0 0 0 - Take 1 from the thousands digit. There are 0 thousands, so we swap 1 from the next column for 10 thousands. If that next digit is also 0 we swap again. Then take 1 away.0 thousands → swap, then 10 − 1 = 9 thousands
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 100,000 1 0 0 0 0 0 99,000 9 9 0 0 0
100,000 − 1,000 = 99,000
Check. Do the opposite: 99,000 + 1,000 = 100,000. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 10,000 less, change the ten thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the ten thousands digit. 10,000 is one ten thousand, so we change the ten thousands digit.
Thousands Ones HTh TTh Th H T O 2 0 5 0 0 0 - Take 1 from the ten thousands digit. There are 0 ten thousands, so we swap 1 from the next column for 10 ten thousands. If that next digit is also 0 we swap again. Then take 1 away.0 ten thousands → swap, then 10 − 1 = 9 ten thousands
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 205,000 2 0 5 0 0 0 195,000 1 9 5 0 0 0
205,000 − 10,000 = 195,000
Check. Do the opposite: 195,000 + 10,000 = 205,000. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 100,000 less, change the hundred thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the hundred thousands digit. 100,000 is one hundred thousand, so we change the hundred thousands digit.
Thousands Ones HTh TTh Th H T O 6 4 0 0 0 0 - Take 1 from the hundred thousands digit. 100,000 less means 1 less hundred thousand.6 − 1 = 5 hundred thousands
- Write the new number. The digits that did not change stay the same.640,000 − 100,000 = 540,000
Check. Do the opposite: 540,000 + 100,000 = 640,000. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 1,000 less, change the thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the thousands digit. 1,000 is one thousand, so we change the thousands digit.
Thousands Ones HTh TTh Th H T O 7 0 0 5 0 0 - Take 1 from the thousands digit. There are 0 thousands, so we swap 1 from the next column for 10 thousands. If that next digit is also 0 we swap again. Then take 1 away.0 thousands → swap, then 10 − 1 = 9 thousands
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 700,500 7 0 0 5 0 0 699,500 6 9 9 5 0 0
700,500 − 1,000 = 699,500
Check. Do the opposite: 699,500 + 1,000 = 700,500. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.
- Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.From 7,260 to 7,270 is 1 jump.
The difference is 10. - Say which way and how big each jump is. The numbers are getting bigger, so we count on.Each jump is +10.
- Fill in the missing numbers. Add (or take away) one jump from a number you know. Where two gaps are side by side, fill one first and use it for the next.Number 1: 7,260 − 10 = 7,250
Number 4: 7,270 + 10 = 7,280 - Write the missing numbers. Put them in the order they appear.7,250, 7,280
Check. Read the whole sequence again and check every jump is +10: 7,250, 7,260, 7,270, 7,280, 7,290. ✓
Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.
Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.
- Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.From 850,000 to 550,000 is 3 jumps.
The difference is 300,000.
300,000 ÷ 3 = 100,000 - Say which way and how big each jump is. The numbers are getting smaller, so we count back.Each jump is −100,000.
- Fill in the missing numbers. Add (or take away) one jump from a number you know. Where two gaps are side by side, fill one first and use it for the next.Number 2: 850,000 − 100,000 = 750,000
Number 3: 750,000 − 100,000 = 650,000 - Write the missing numbers. Put them in the order they appear.750,000, 650,000
Check. Read the whole sequence again and check every jump is −100,000: 850,000, 750,000, 650,000, 550,000, 450,000. ✓
Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.
Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.
- Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.From 395,000 to 397,000 is 2 jumps.
The difference is 2,000.
2,000 ÷ 2 = 1,000 - Say which way and how big each jump is. The numbers are getting bigger, so we count on.Each jump is +1,000.
- Fill in the missing numbers. Add (or take away) one jump from a number you know. Where two gaps are side by side, fill one first and use it for the next.Number 2: 395,000 + 1,000 = 396,000
Number 5: 398,000 + 1,000 = 399,000
Number 6: 399,000 + 1,000 = 400,000 - Write the missing numbers. Put them in the order they appear.396,000, 399,000, 400,000
Check. Read the whole sequence again and check every jump is +1,000: 395,000, 396,000, 397,000, 398,000, 399,000, 400,000. ✓
Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.
Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.
- Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.From 600,000 to 800,000 is 2 jumps.
The difference is 200,000.
200,000 ÷ 2 = 100,000 - Say which way and how big each jump is. The numbers are getting bigger, so we count on.Each jump is +100,000.
- Fill in the missing numbers. Add (or take away) one jump from a number you know. Where two gaps are side by side, fill one first and use it for the next.Number 2: 600,000 + 100,000 = 700,000
Number 4: 800,000 + 100,000 = 900,000 - Write the missing numbers. Put them in the order they appear.700,000, 900,000
Check. Read the whole sequence again and check every jump is +100,000: 600,000, 700,000, 800,000, 900,000, 1,000,000. ✓
Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.
Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.
- Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.From 4,000 to 3,700 is 3 jumps.
The difference is 300.
300 ÷ 3 = 100 - Say which way and how big each jump is. The numbers are getting smaller, so we count back.Each jump is −100.
- Fill in the missing numbers. Add (or take away) one jump from a number you know. Where two gaps are side by side, fill one first and use it for the next.Number 2: 4,000 − 100 = 3,900
Number 3: 3,900 − 100 = 3,800 - Write the missing numbers. Put them in the order they appear.3,900, 3,800
Check. Read the whole sequence again and check every jump is −100: 4,000, 3,900, 3,800, 3,700. ✓
Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.
Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.
- Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.From 240,000 to 270,000 is 3 jumps.
The difference is 30,000.
30,000 ÷ 3 = 10,000 - Say which way and how big each jump is. The numbers are getting bigger, so we count on.Each jump is +10,000.
- Fill in the missing numbers. Add (or take away) one jump from a number you know. Where two gaps are side by side, fill one first and use it for the next.Number 2: 240,000 + 10,000 = 250,000
Number 3: 250,000 + 10,000 = 260,000
Number 6: 280,000 + 10,000 = 290,000
Number 8: 300,000 + 10,000 = 310,000 - Write the missing numbers. Put them in the order they appear.250,000, 260,000, 290,000, 310,000
Check. Read the whole sequence again and check every jump is +10,000: 240,000, 250,000, 260,000, 270,000, 280,000, 290,000, 300,000, 310,000, 320,000. ✓
Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.
Method. Work out how much each term changes by. If the jump is always the same, that is the rule.
- Write the first terms in a place-value table. Look for the one column that changes.
Thousands Ones Th H T O 5,320 5 3 2 0 5,420 5 4 2 0 5,520 5 5 2 0 - Find the jump between two terms next to each other. The column that changes tells us the size of the jump.5,420 − 5,320 = 100 (it goes up)
- Check the jump is the same all the way along. A rule has to work for every pair of terms.5,420 − 5,320 = 100
5,520 − 5,420 = 100
5,620 − 5,520 = 100 - Write the rule and the next two terms. Add 100 each time.Rule: add 100.
Next two terms: 5,720, 5,820
Check. 5,620 + 100 = 5,720 and 5,720 + 100 = 5,820. ✓
Watch out. Check more than one pair of terms. One pair is not enough to be sure of the rule.
Method. Work out how much each term changes by. If the jump is always the same, that is the rule.
- Write the first terms in a place-value table. Look for the one column that changes.
Thousands Ones HTh TTh Th H T O 800,000 8 0 0 0 0 0 700,000 7 0 0 0 0 0 600,000 6 0 0 0 0 0 - Find the jump between two terms next to each other. The column that changes tells us the size of the jump.800,000 − 700,000 = 100,000 (it goes down)
- Check the jump is the same all the way along. A rule has to work for every pair of terms.800,000 − 700,000 = 100,000
700,000 − 600,000 = 100,000
600,000 − 500,000 = 100,000 - Write the rule and the next two terms. Subtract 100,000 each time.Rule: subtract 100,000.
Next two terms: 400,000, 300,000
Check. 500,000 − 100,000 = 400,000 and 400,000 − 100,000 = 300,000. ✓
Watch out. Check more than one pair of terms. One pair is not enough to be sure of the rule.
Method. Work out how much each term changes by. If the jump is always the same, that is the rule.
- Write the first terms in a place-value table. Look for the one column that changes.
Thousands Ones HTh TTh Th H T O 204,000 2 0 4 0 0 0 214,000 2 1 4 0 0 0 224,000 2 2 4 0 0 0 - Find the jump between two terms next to each other. The column that changes tells us the size of the jump.214,000 − 204,000 = 10,000 (it goes up)
- Check the jump is the same all the way along. A rule has to work for every pair of terms.214,000 − 204,000 = 10,000
224,000 − 214,000 = 10,000
234,000 − 224,000 = 10,000 - Write the rule and the next two terms. Add 10,000 each time.Rule: add 10,000.
Next two terms: 244,000, 254,000
Check. 234,000 + 10,000 = 244,000 and 244,000 + 10,000 = 254,000. ✓
Watch out. Check more than one pair of terms. One pair is not enough to be sure of the rule.
Method. Work out how much each term changes by. If the jump is always the same, that is the rule.
- Write the first terms in a place-value table. Look for the one column that changes.
Thousands Ones Th H T O 8,090 8 0 9 0 8,080 8 0 8 0 8,070 8 0 7 0 - Find the jump between two terms next to each other. The column that changes tells us the size of the jump.8,090 − 8,080 = 10 (it goes down)
- Check the jump is the same all the way along. A rule has to work for every pair of terms.8,090 − 8,080 = 10
8,080 − 8,070 = 10
8,070 − 8,060 = 10 - Write the rule and the next two terms. Subtract 10 each time.Rule: subtract 10.
Next two terms: 8,050, 8,040
Check. 8,060 − 10 = 8,050 and 8,050 − 10 = 8,040. ✓
Watch out. Check more than one pair of terms. One pair is not enough to be sure of the rule.
Method. Look at the two numbers in the number sentence. Find which column changes, or use the opposite (inverse) operation, to find the missing number.
- Look at the number sentence. The box stands for the number we want.245,600 + ? = 345,600
- Put the two known numbers in a place-value table. Only one column changes, and it shows the size of the jump.
Thousands Ones HTh TTh Th H T O 245,600 2 4 5 6 0 0 345,600 3 4 5 6 0 0 - Work out the missing number. Use the opposite operation to check the size of the change.345,600 − 245,600 = 100,000
- Check it in the number sentence. Put the answer back in the box.245,600 + 100,000 = 345,600 ✓
Check. The change is 100,000, a power of 10 (or a number made from powers of 10). ✓
Watch out. The box is not always at the start. Think about what the sentence is saying before you choose + or −.
Method. Look at the two numbers in the number sentence. Find which column changes, or use the opposite (inverse) operation, to find the missing number.
- Look at the number sentence. The box stands for the number we want.? − 1,000 = 87,500
- Put the two known numbers in a place-value table. Only one column changes, and it shows the size of the jump.
Thousands Ones TTh Th H T O 87,500 8 7 5 0 0 88,500 8 8 5 0 0 - Work out the missing number. Use the opposite operation to check the size of the change.87,500 + 1,000 = 88,500
- Check it in the number sentence. Put the answer back in the box.88,500 − 1,000 = 87,500 ✓
Check. The change is 1,000, a power of 10 (or a number made from powers of 10). ✓
Watch out. The box is not always at the start. Think about what the sentence is saying before you choose + or −.
Method. Look at the two numbers in the number sentence. Find which column changes, or use the opposite (inverse) operation, to find the missing number.
- Look at the number sentence. The box stands for the number we want.630,000 − ? = 620,000
- Put the two known numbers in a place-value table. Only one column changes, and it shows the size of the jump.
Thousands Ones HTh TTh Th H T O 620,000 6 2 0 0 0 0 630,000 6 3 0 0 0 0 - Work out the missing number. Use the opposite operation to check the size of the change.630,000 − 620,000 = 10,000
- Check it in the number sentence. Put the answer back in the box.630,000 − 10,000 = 620,000 ✓
Check. The change is 10,000, a power of 10 (or a number made from powers of 10). ✓
Watch out. The box is not always at the start. Think about what the sentence is saying before you choose + or −.
Method. Look at the two numbers in the number sentence. Find which column changes, or use the opposite (inverse) operation, to find the missing number.
- Look at the number sentence. The box stands for the number we want.? + 100 = 4,000
- Put the two known numbers in a place-value table. Only one column changes, and it shows the size of the jump.
Thousands Ones Th H T O 3,900 3 9 0 0 4,000 4 0 0 0 - Work out the missing number. Use the opposite operation to check the size of the change.4,000 − 100 = 3,900
- Check it in the number sentence. Put the answer back in the box.3,900 + 100 = 4,000 ✓
Check. The change is 100, a power of 10 (or a number made from powers of 10). ✓
Watch out. The box is not always at the start. Think about what the sentence is saying before you choose + or −.
Method. Add 10,000 for each day. Stop when the number is more than 100,000.
- Start with the views now. This is day 0.Day 0: 48,500
- Add 10,000 each day. Only the ten thousands digit changes (and a carry when it passes 9).Day 1: 48,500 + 10,000 = 58,500
Day 2: 58,500 + 10,000 = 68,500
Day 3: 68,500 + 10,000 = 78,500
Day 4: 78,500 + 10,000 = 88,500
Day 5: 88,500 + 10,000 = 98,500
Day 6: 98,500 + 10,000 = 108,500 - Find the first number over 100,000. Day 5 has 98,500, which is still less than 100,000. Day 6 has 108,500.Day 6
Check. 48,500 + 6 × 10,000 = 108,500, which is more than 100,000. ✓
Watch out. The number has to be more than 100,000. Make sure you do not stop at 98,500 (day 5).
Method. Find the total distance, then work out how many jumps of 10,000 fit into it.
- Find how far it is from the start to the end. Subtract the start from the end.200,000 − 120,000 = 80,000
- Divide by the size of each jump. Each jump is 10,000.80,000 ÷ 10,000 = 8
- Write the answer. 8 jumps of 10,000
Check. 8 × 10,000 = 80,000 and 120,000 + 80,000 = 200,000. ✓
Watch out. Count the jumps, not the numbers you say. Counting the numbers would include the start and give one too many.
Method. Find the total distance, then work out how many jumps of 100 fit into it.
- Find how far it is from the start to the end. Subtract the start from the end.4,200 − 3,500 = 700
- Divide by the size of each jump. Each jump is 100.700 ÷ 100 = 7
- Write the answer. 7 jumps of 100
Check. 7 × 100 = 700 and 3,500 + 700 = 4,200. ✓
Watch out. Count the jumps, not the numbers you say. Counting the numbers would include the start and give one too many.
Method. Find the total distance, then work out how many jumps of 100,000 fit into it.
- Find how far it is from the start to the end. Subtract the start from the end.650,000 − 150,000 = 500,000
- Divide by the size of each jump. Each jump is 100,000.500,000 ÷ 100,000 = 5
- Write the answer. 5 jumps of 100,000
Check. 5 × 100,000 = 500,000 and 150,000 + 500,000 = 650,000. ✓
Watch out. Count the jumps, not the numbers you say. Counting the numbers would include the start and give one too many.
Method. To find 10 more, change the tens digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the tens digit. 10 is one ten, so we change the tens digit.
Thousands Ones TTh Th H T O 9 9 9 9 5 - Add 1 to the tens digit. There are 9 tens. 1 more ten makes 10 tens, which is 1 of the next column and 0 tens. We carry (and the carry carries on if the next digit is also 9).9 + 1 = 10 → write 0 in the T column and carry 1
- Write the new number. The digits that did not change stay the same.
Thousands Ones HTh TTh Th H T O 99,995 9 9 9 9 5 100,005 1 0 0 0 0 5
99,995 + 10 = 100,005
Check. Do the opposite: 100,005 − 10 = 99,995. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. To find 100 more, change the hundreds digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the hundreds digit. 100 is one hundred, so we change the hundreds digit.
Thousands Ones TTh Th H T O 4 9 9 5 0 - Add 1 to the hundreds digit. There are 9 hundreds. 1 more hundred makes 10 hundreds, which is 1 of the next column and 0 hundreds. We carry (and the carry carries on if the next digit is also 9).9 + 1 = 10 → write 0 in the H column and carry 1
- Write the new number. The digits that did not change stay the same.
Thousands Ones TTh Th H T O 49,950 4 9 9 5 0 50,050 5 0 0 5 0
49,950 + 100 = 50,050
Check. Do the opposite: 50,050 − 100 = 49,950. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. Count back in 1,000s, one minute at a time. Each minute is one jump.
- Say what each jump is. Each minute, the amount goes down by 1,000.1 minute = −1,000
- Count 7 jumps. Start each jump from the number you have just found.Minute 1: 25,000 − 1,000 = 24,000
Minute 2: 24,000 − 1,000 = 23,000
Minute 3: 23,000 − 1,000 = 22,000
Minute 4: 22,000 − 1,000 = 21,000
Minute 5: 21,000 − 1,000 = 20,000
Minute 6: 20,000 − 1,000 = 19,000
Minute 7: 19,000 − 1,000 = 18,000 - Write the answer. 18,000 litres
Check. 7 × 1,000 = 7,000 and 25,000 − 7,000 = 18,000. ✓
Watch out. Count the jumps carefully. Tally marks can help you keep track.
Method. The tank is empty when all 25,000 litres have gone. Find how many lots of 1,000 litres are in 25,000.
- Find the amount to drain. The whole tank.25,000 litres
- Divide by the amount that drains each minute. Each minute is 1,000 litres.25,000 ÷ 1,000 = 25
- Write the answer. 25 minutes
Check. 25 × 1,000 = 25,000. ✓
Watch out. Count the minutes, not the litres.
Method. Work out 1,000 less than 300,000 yourself, then compare it with Sam’s answer.
- Put 300,000 in a place-value table. The thousands digit is 0.
Thousands Ones HTh TTh Th H T O 3 0 0 0 0 0 - Take 1,000 away. There are 0 thousands, so we swap 1 hundred thousand for 10 ten thousands, then 1 ten thousand for 10 thousands. Then take 1 thousand away.300,000 − 1,000 = 299,000
- Say what Sam did. Sam wrote 200,000. That is 300,000 − 100,000, so he took away 100,000 instead of 1,000.He changed the hundred thousands digit instead of the thousands digit.
Check. 299,000 + 1,000 = 300,000. ✓
Watch out. Change the digit in the column that matches the step: 1,000 means the thousands column.
“When you add 1,000 to a number, only the thousands digit changes.”
Method. Try adding 1,000 to some numbers. Look at which digits change.
- Try 342,150. The thousands digit is 2.342,150 + 1,000 = 343,150. Only the thousands digit changed. It works.
- Try 349,150. The thousands digit is 9, so we carry.349,150 + 1,000 = 350,150. The thousands, ten thousands digits changed. It does not work.
- Write the answer. It works when the thousands digit is not 9.Sometimes
Check. It only fails when the thousands digit is 9. ✓
Watch out. 9s are the problem: adding to a 9 carries to the next column.
Method. Write out the numbers she says and see whether 100,000 is one of them.
- Count on in 10,000s from 5,000. Add 10,000 each time.5,000, 15,000, 25,000, 35,000, 45,000, 55,000, 65,000, 75,000, 85,000, 95,000, 105,000
- Look for 100,000. She goes from 95,000 to 105,000, so she jumps over it.95,000 < 100,000 < 105,000
- Explain why. Every number she says ends in 5,000, but 100,000 ends in 0,000.No, she will not say 100,000.
Check. 100,000 − 5,000 = 95,000, and 95,000 ÷ 10,000 = 9.5, which is not a whole number of jumps. ✓
Watch out. A number is only in the sequence if you can reach it with a whole number of jumps.
Method. Find how many jumps of 1,000 are needed to get past 50,000. Then work out the number.
- Find how far 7,125 is from 50,000. Subtract.50,000 − 7,125 = 42,875
- Divide by the size of each jump. This tells us how many whole jumps fit before 50,000.42,875 ÷ 1,000 = 42 remainder 875
So 42 jumps are not enough (we land on 49,125). We need 43 jumps. - Count 43 jumps. The last few numbers show it going over.47,125, 48,125, 49,125, 50,125
- Write the answer. 50,125 after 43 jumps
Check. 7,125 + 43 × 1,000 = 50,125, which is more than 50,000. ✓
Watch out. The number must be more than 50,000.