Year 6: Comparing and ordering numbers
Part of the Year 6 place value and negative numbers chapter. Chapter overview · whole chapter on one page
Comparing and ordering numbers
To compare two numbers, start at the left. Look at the biggest columns first. The first place where the digits differ tells you which number is bigger. You do not need to look at the other digits at all.
A number with more digits is bigger, as long as it does not start with a zero. So 1,000,000 (seven digits) is bigger than 999,999 (six digits). Use the signs < and > to show the order. The open end always points to the bigger number.
| Millions | Thousands | Ones | ||||||
|---|---|---|---|---|---|---|---|---|
| Number | TM | M | HTh | TTh | Th | H | T | O |
| 4,628,100 | 4 | 6 | 2 | 8 | 1 | 0 | 0 | |
| 4,618,900 | 4 | 6 | 1 | 8 | 9 | 0 | 0 | |
Ordering. To put a list in order, compare the numbers two at a time, or find the biggest and smallest first. Write the smallest number first when you order from the smallest. Then check that each number is bigger than the one before.
Digit cards. Sometimes you are given digits and asked to make the largest or smallest number. Use each digit once. Put the biggest digit on the left for the largest number. For the smallest, you cannot start with zero, so use the smallest digit that is not zero first.
Key ideas
- Compare the digits from the left. The first column where the digits differ decides.
- More digits means a bigger number, if neither number starts with a zero.
- The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
- For the largest number, put the biggest digit in the millions column. For the smallest, do not start with 0.
- Check an order by reading it again: each number should be bigger than the one before.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the digits from the left. The first place where they differ decides.
- Line the numbers up and compare from the left. The first column where the digits differ decides which number is bigger.4,628,100 and 4,618,900 both have 7 digits.
- Look at the ten thousands column. The digits are 2 and 1. The larger digit means the larger number.2 and 1, so 4,628,100 is bigger
- Choose the sign. The open end of the sign faces the bigger number.4,628,100 > 4,618,900
Check. Compare the other way round: 4,618,900 and 4,628,100 give the opposite sign. ✓
Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the numbers in pairs, then write them from the smallest to the other.
- Compare the numbers. Look at the number of digits, then compare the digits from the left.3,040,000, 3,400,000, 304,000, 3,000,450
- Write the numbers from the smallest. Put the smallest number first.304,000, 3,000,450, 3,040,000, 3,400,000
- Check the order. Each number should be bigger than the one before it.304,000 < 3,000,450 < 3,040,000 < 3,400,000
Check. The first number is the smallest and the last number is the largest. ✓
Watch out. Do not order by the first digit alone if the numbers have different numbers of digits.
Method. Put the biggest digit in the leftmost column, then the next biggest, and so on.
- Sort the digits from biggest to smallest. The leftmost column has the biggest value.9 7 5 3 2 1 0
- Write the digits in that order. Each digit goes in the next column to the right.9,753,210
- Check that every digit is used once. Count the digits in your answer.7 digits, and the digits are the same as the cards
Check. Every digit is used once: 9,753,210 has 7 digits. ✓
Watch out. The biggest digit must go in the biggest column.
Watch out
Do not compare the last digits first. 4,999,999 is less than 5,000,000 even though it has more 9s. Look at the first digit that is different: here it is the 4 and the 5. Always count the digits first, so that you do not place a number with fewer digits above one with more digits.Exercise 1B Comparing and ordering
Non-calculator| Millions | Thousands | Ones | ||||||
|---|---|---|---|---|---|---|---|---|
| Number | TM | M | HTh | TTh | Th | H | T | O |
| 6,250,000 | 6 | 2 | 5 | 0 | 0 | 0 | 0 | |
| 6,205,000 | 6 | 2 | 0 | 5 | 0 | 0 | 0 | |
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the digits from the left. The first place where they differ decides.
- Line the numbers up and compare from the left. The first column where the digits differ decides which number is bigger.6,250,000 and 6,205,000 both have 7 digits.
- Look at the ten thousands column. The digits are 5 and 0. The larger digit means the larger number.5 and 0, so 6,250,000 is bigger
- Choose the sign. The open end of the sign faces the bigger number.6,250,000 > 6,205,000
Check. Compare the other way round: 6,205,000 and 6,250,000 give the opposite sign. ✓
Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the digits from the left. The first place where they differ decides.
- Line the numbers up and compare from the left. The first column where the digits differ decides which number is bigger.3,409,000 and 3,490,000 both have 7 digits.
- Look at the ten thousands column. The digits are 0 and 9. The larger digit means the larger number.0 and 9, so 3,490,000 is bigger
- Choose the sign. The open end of the sign faces the bigger number.3,409,000 < 3,490,000
Check. Compare the other way round: 3,490,000 and 3,409,000 give the opposite sign. ✓
Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the digits from the left. The first place where they differ decides.
- Line the numbers up and compare from the left. The first column where the digits differ decides which number is bigger.1,008,000 and 1,080,000 both have 7 digits.
- Look at the ten thousands column. The digits are 0 and 8. The larger digit means the larger number.0 and 8, so 1,080,000 is bigger
- Choose the sign. The open end of the sign faces the bigger number.1,008,000 < 1,080,000
Check. Compare the other way round: 1,080,000 and 1,008,000 give the opposite sign. ✓
Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the digits from the left. The first place where they differ decides.
- Count the digits in each number. When neither number starts with a zero, the number with more digits is bigger.950,000 has 6 digits. 1,000,000 has 7 digits.
- Choose the sign. The open end of the sign faces the bigger number.950,000 < 1,000,000
Check. Compare the other way round: 1,000,000 and 950,000 give the opposite sign. ✓
Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the digits from the left. The first place where they differ decides.
- Count the digits in each number. When neither number starts with a zero, the number with more digits is bigger.4,305,000 has 7 digits. 435,000 has 6 digits.
- Choose the sign. The open end of the sign faces the bigger number.4,305,000 > 435,000
Check. Compare the other way round: 435,000 and 4,305,000 give the opposite sign. ✓
Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the digits from the left. The first place where they differ decides.
- Line the numbers up and compare from the left. The first column where the digits differ decides which number is bigger.2,999,999 and 3,000,000 both have 7 digits.
- Look at the millions column. The digits are 2 and 3. The larger digit means the larger number.2 and 3, so 3,000,000 is bigger
- Choose the sign. The open end of the sign faces the bigger number.2,999,999 < 3,000,000
Check. Compare the other way round: 3,000,000 and 2,999,999 give the opposite sign. ✓
Watch out. Do not compare the last digits first. 4,999,999 is less than 5,000,000, even though it has more 9s.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the numbers in pairs, then write them from the smallest to the other.
- Compare the numbers. Look at the number of digits, then compare the digits from the left.2,409,000, 240,900, 2,490,000, 2,040,900, 2,900,400
- Write the numbers from the smallest. Put the smallest number first.240,900, 2,040,900, 2,409,000, 2,490,000, 2,900,400
- Check the order. Each number should be bigger than the one before it.240,900 < 2,040,900 < 2,409,000 < 2,490,000 < 2,900,400
Check. The first number is the smallest and the last number is the largest. ✓
Watch out. Do not order by the first digit alone if the numbers have different numbers of digits.
Key words. To compare two numbers, use the signs < (is less than) and > (is greater than). The open end of the sign faces the bigger number: 9 > 4 and 4 < 9.
Method. Compare the numbers in pairs, then write them from the largest to the other.
- Compare the numbers. Look at the number of digits, then compare the digits from the left.5,000,001, 5,100,000, 4,999,999, 5,010,000
- Write the numbers from the largest. Put the largest number first.5,100,000, 5,010,000, 5,000,001, 4,999,999
- Check the order. Each number should be smaller than the one before it.5,100,000 > 5,010,000 > 5,000,001 > 4,999,999
Check. The first number is the largest and the last number is the smallest. ✓
Watch out. Do not order by the first digit alone if the numbers have different numbers of digits.
Method. Put the biggest digit in the leftmost column, then the next biggest, and so on.
- Sort the digits from biggest to smallest. The leftmost column has the biggest value.9 8 6 4 2 1 0
- Write the digits in that order. Each digit goes in the next column to the right.9,864,210
- Check that every digit is used once. Count the digits in your answer.7 digits, and the digits are the same as the cards
Check. Every digit is used once: 9,864,210 has 7 digits. ✓
Watch out. The biggest digit must go in the biggest column.
Method. The first digit cannot be zero. Use the smallest digit that is not zero, then the rest in order from smallest.
- Choose the first digit. A number cannot start with a zero, so the first digit must be a digit that is not zero.The smallest digit that is not zero is 1.
- Write the other digits from smallest to biggest. The smaller digits go in the columns on the right.1 then 0 2 4 6 8 9
- Write the number. 1,024,689
Check. The number has 7 digits, and every digit is used once. ✓
Watch out. Zero can be used, but not as the first digit.
Method. Work out how far each number is from 5,000,000. The smallest distance is the closest number.
- Find each distance from 5,000,000. Subtract the smaller number from the bigger one.
Number Distance from 5,000,000 4,820,000 180,000 5,060,000 60,000 4,990,000 10,000 closest 5,400,000 400,000 - Choose the smallest distance. The closest number has the smallest distance.The smallest distance is 10,000, so the answer is option C: 4,990,000
Check. 4,990,000 is 10,000 away from 5,000,000. The next closest is 5,060,000, which is 60,000 away. ✓
Watch out. Do not choose the number that is closest to the start of the list. Compare the distances.
Method. Check each part: the number must be bigger than 3,000,000 and smaller than 4,000,000.
- Check 3,090,000. It is more than 3,000,000 and less than 4,000,000.3,000,000 < 3,090,000 < 4,000,000 ✓
- Check 4,010,000. It is more than 4,000,000, so it is not in the range.4,010,000 > 4,000,000 ✗
- Check 2,999,999 and 3,999,999. 2,999,999 is less than 3,000,000. 3,999,999 is between the two numbers.2,999,999 < 3,000,000 ✗; 3,999,999 is in range ✓
- Write the numbers in the range. 3,090,000 and 3,999,999
Check. Both answers are between 3,000,000 and 4,000,000. ✓
Watch out. Check both ends of the range. 3,999,999 is still less than 4,000,000.
Method. Halfway means the number in the middle, so find the gap and halve it.
- Find the gap. The gap is the difference between the two numbers.5,000,000 − 4,000,000 = 1,000,000
- Halve the gap. Halving gives the distance from either end to the middle.1,000,000 ÷ 2 = 500,000
- Add the half gap to the start. Add it to the smaller number.4,000,000 + 500,000 = 4,500,000
Check. 4,500,000 is the same distance from 4,000,000 and from 5,000,000. ✓
Watch out. Halfway is not the same as adding 1,000,000.
Method. The largest 6-digit number is 999,999. Add one to it.
- Find the largest 6-digit number. Six digits, all nines.999,999
- Add 1. Adding 1 makes the next number, which has seven digits.999,999 + 1 = 1,000,000
Check. 1,000,000 has seven digits, so it is bigger than any 6-digit number. ✓
Watch out. When you add 1 to a row of nines, every nine turns into a zero.
Method. A number with seven digits starts at 1,000,000. A number with six digits stops at 999,999.
- Find the smallest 7-digit number. A 7-digit number has seven columns, so the first digit is at least 1 in the millions column.1,000,000
- Find the largest 6-digit number. A 6-digit number has six columns, so it stops before 1,000,000.999,999
- Compare them. Every 7-digit number is at least 1,000,000, which is bigger than 999,999.1,000,000 > 999,999, so 7-digit numbers are always bigger
Check. Try any 7-digit number, such as 1,250,000, and any 6-digit number, such as 987,654. The 7-digit number is always bigger. ✓
Watch out. Always needs a reason that works for every case. One counter-example would make it false.
Method. The smallest 7-digit number with no zeros is 1,111,111. Add 1 to it.
- Find the smallest 7-digit number with no zeros. The smallest digit that is not zero is 1, so use 1 in every column.1,111,111
- Add 1. Adding 1 changes only the ones digit.1,111,111 + 1 = 1,111,112
Check. 1,111,112 has seven digits and no zeros. ✓
Watch out. Zero is not allowed in this puzzle, so 1,000,000 does not count.