Year 5: Comparing and ordering numbers to 1,000,000
Part of the Year 5 place value to 1,000,000 chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Comparing and ordering numbers to 1,000,000
Comparing. To decide which number is bigger, first look at how many digits each number has. A number with more digits is bigger (as long as it does not start with 0). If both have the same number of digits, line the digits up in a place-value table and compare from the left. The first column that is different decides.
| Thousands | Ones | |||||
|---|---|---|---|---|---|---|
| HTh | TTh | Th | H | T | O | |
| 482,619 | 4 | 8 | 2 | 6 | 1 | 9 |
| 482,691 | 4 | 8 | 2 | 6 | 9 | 1 |
We use three signs: < means “is less than”, > means “is greater than” and = means “is equal to”. The open end of the sign faces the bigger number. To order numbers, compare them in pairs. Ascending order is smallest first. Descending order is biggest first.
Number lines. To read a number line, work out what one gap is worth: take the difference between two labelled numbers and divide by the number of gaps. This line goes from 0 to 1,000,000 in 10 gaps, so each gap is 1,000,000 ÷ 10 = 100,000. The arrow is between 600,000 and 700,000. If the arrow is not exactly on a mark we estimate a number close to it. Halfway between 600,000 and 700,000 is 650,000.
Making numbers with digit cards. To make the largest number, put the biggest digit in the biggest column, then the next biggest, and so on. To make the smallest number, put the smallest digit first, but never put 0 in the first column. A six-digit number cannot start with 0.
Key words. Ascending means getting bigger. Descending means getting smaller. To estimate is to give a sensible answer that is close to the exact one.
Key ideas
- More digits means a bigger number (if no number starts with 0).
- Same number of digits: compare from the left. The first column that is different decides.
- < less than, > greater than, = equal to. The open end faces the bigger number.
- On a number line, work out what one gap is worth before you read the arrow.
- Largest number: biggest digit first. Smallest number: smallest digit first, but not 0.
- Halfway between two numbers: add them and halve the total.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 356,742 3 5 6 7 4 2 356,724 3 5 6 7 2 4 - Compare the digits from the left. The first column where they are different decides which number is bigger.hundred thousands: 3 and 3 are the same, so look at the next column.
ten thousands: 5 and 5 are the same, so look at the next column.
thousands: 6 and 6 are the same, so look at the next column.
hundreds: 7 and 7 are the same, so look at the next column.
tens: 4 and 2 are different. 4 > 2, so 356,742 is bigger. - Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.356,742 > 356,724
Check. Picture a number line: 356,724 is to the left of 356,742. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 98,765 9 8 7 6 5 100,234 1 0 0 2 3 4 - Count the digits. A number with more digits is bigger (when no number starts with 0).100,234 has 6 digits and 98,765 has 5 digits, so 100,234 is bigger.
- Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.98,765 < 100,234
Check. Picture a number line: 98,765 is to the left of 100,234. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write each side as one number. We can only compare numbers when they are written the usual way.300,000 + 4,000 = 304,000
340,000 = 340,000 - Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 304,000 3 0 4 0 0 0 340,000 3 4 0 0 0 0 - Compare the digits from the left. The first column where they are different decides which number is bigger.hundred thousands: 3 and 3 are the same, so look at the next column.
ten thousands: 0 and 4 are different. 0 < 4, so 304,000 is smaller. - Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.304,000 < 340,000
Check. Picture a number line: 304,000 is to the left of 340,000. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. Ascending order means smallest first. Descending order means biggest first.
Method. Put the numbers in a place-value table, then compare columns from the left. Smallest first.
- Write all the numbers in a place-value table. This lines up the digits so we can compare them.
Thousands Ones HTh TTh Th H T O 345,678 3 4 5 6 7 8 345,768 3 4 5 7 6 8 354,678 3 5 4 6 7 8 345,687 3 4 5 6 8 7 34,568 3 4 5 6 8 - Compare the numbers in pairs, from the left. Where the digits in a column match, move on to the next column.34,568 and 345,678: 345,678 has 6 digits and 34,568 has 5 digits, so 345,678 is bigger.
345,678 and 345,687: tens: 7 and 8 are different. 7 < 8, so 345,678 is smaller.
345,687 and 345,768: hundreds: 6 and 7 are different. 6 < 7, so 345,687 is smaller.
345,768 and 354,678: ten thousands: 4 and 5 are different. 4 < 5, so 345,768 is smaller. - Write them smallest first. Check you have used every number once.34,568, 345,678, 345,687, 345,768, 354,678
Check. Count the numbers: 5 in the question and 5 in the answer. ✓
Watch out. Tick each number off as you write it, so you do not miss one.
Key words. Ascending order means smallest first. Descending order means biggest first.
Method. Put the numbers in a place-value table, then compare columns from the left. Biggest first.
- Write all the numbers in a place-value table. This lines up the digits so we can compare them.
Thousands Ones HTh TTh Th H T O 807,060 8 0 7 0 6 0 870,600 8 7 0 6 0 0 806,070 8 0 6 0 7 0 87,060 8 7 0 6 0 - Compare the numbers in pairs, from the left. Where the digits in a column match, move on to the next column.87,060 and 806,070: 806,070 has 6 digits and 87,060 has 5 digits, so 806,070 is bigger.
806,070 and 807,060: thousands: 6 and 7 are different. 6 < 7, so 806,070 is smaller.
807,060 and 870,600: ten thousands: 0 and 7 are different. 0 < 7, so 807,060 is smaller. - Write them biggest first. Check you have used every number once.870,600, 807,060, 806,070, 87,060
Check. Count the numbers: 4 in the question and 4 in the answer. ✓
Watch out. Tick each number off as you write it, so you do not miss one.
Method. Find out what each small mark is worth, then count on from a number you can read.
- Count the gaps between two labelled numbers. The line goes from 300,000 to 400,000 in 10 equal gaps.400,000 − 300,000 = 100,000
- Work out what one gap is worth. Share the distance equally between the gaps.100,000 ÷ 10 = 10,000
- Count on from the start to the arrow. The arrow is 6 gaps after 300,000.300,000 + 6 × 10,000 = 300,000 + 60,000 = 360,000
Check. Count back from 400,000: 4 gaps back is 400,000 − 40,000 = 360,000. ✓
Watch out. Do not read the nearest label. Count the small marks to find the exact number.
Method. Work out what each gap is worth. Then decide which two gaps the arrow is between and how far along it is.
- Work out what one gap is worth. The line goes from 0 to 1,000,000 in 10 equal gaps.1,000,000 ÷ 10 = 100,000
- Find the two numbers either side of the arrow. The arrow is after 6 gaps, but before 7 gaps.600,000 < arrow < 700,000
- Decide how far along the arrow is. Halfway between 600,000 and 700,000 is 650,000.The arrow is close to 650,000.
- Choose the best estimate. Pick the option nearest to the arrow.C (650,000)
Check. The other options are far away: 150,000, 450,000, 950,000. ✓
Watch out. An estimate is a sensible guess. You do not need the exact number.
Method. Make the largest by putting the cards in order, biggest first. Make the smallest by putting them in order, smallest first (with a non-zero digit in front).
- Largest: put the digits in order, biggest first. The biggest digit in the biggest column gives the biggest number.9, 7, 5, 4, 2, 0 → 975,420
- Smallest: put the digits in order, smallest first. That would start with 0, but a six-digit number cannot start with 0.0, 2, 4, 5, 7, 9
- Swap the first two digits. The smallest digit that is not 0 goes first, then the 0.2, 0, 4, 5, 7, 9 → 204,579
Check. Check the smallest has six digits and uses all the cards once. 204,579 does. ✓
Watch out. A leading zero is not allowed: 024,579 is really a five-digit number.
Method. The number halfway between two numbers is the same distance from both. Find the gap between them, halve it, then add that to the smaller number.
- Find the gap between the two numbers. Subtract the smaller from the bigger.360,000 − 340,000 = 20,000
- Halve the gap. Halfway is half of the gap away from each number.20,000 ÷ 2 = 10,000
- Add half the gap to the smaller number. This moves us to the middle.340,000 + 10,000 = 350,000
Check. Another way: add the numbers and halve. 340,000 + 360,000 = 700,000 and 700,000 ÷ 2 = 350,000. ✓
Watch out. Halfway is not the average of the digits. Use the whole numbers.
Watch out
Start on the left when you compare, not the right. 98,765 is smaller than 100,234 even though 9 is bigger than 1, because 98,765 has only five digits and 100,234 has six. And a zero cannot go first: the smallest six-digit number is 100,000, not 000,000.Exercise 1B Comparing and ordering
Non-calculatorKey words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 356,742 3 5 6 7 4 2 356,724 3 5 6 7 2 4 - Compare the digits from the left. The first column where they are different decides which number is bigger.hundred thousands: 3 and 3 are the same, so look at the next column.
ten thousands: 5 and 5 are the same, so look at the next column.
thousands: 6 and 6 are the same, so look at the next column.
hundreds: 7 and 7 are the same, so look at the next column.
tens: 4 and 2 are different. 4 > 2, so 356,742 is bigger. - Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.356,742 > 356,724
Check. Picture a number line: 356,724 is to the left of 356,742. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 89,999 8 9 9 9 9 100,001 1 0 0 0 0 1 - Count the digits. A number with more digits is bigger (when no number starts with 0).100,001 has 6 digits and 89,999 has 5 digits, so 100,001 is bigger.
- Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.89,999 < 100,001
Check. Picture a number line: 89,999 is to the left of 100,001. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write each side as one number. We can only compare numbers when they are written the usual way.400,000 + 5,000 = 405,000
450,000 = 450,000 - Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 405,000 4 0 5 0 0 0 450,000 4 5 0 0 0 0 - Compare the digits from the left. The first column where they are different decides which number is bigger.hundred thousands: 4 and 4 are the same, so look at the next column.
ten thousands: 0 and 5 are different. 0 < 5, so 405,000 is smaller. - Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.405,000 < 450,000
Check. Picture a number line: 405,000 is to the left of 450,000. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 620,300 6 2 0 3 0 0 62,030 6 2 0 3 0 - Count the digits. A number with more digits is bigger (when no number starts with 0).620,300 has 6 digits and 62,030 has 5 digits, so 620,300 is bigger.
- Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.620,300 > 62,030
Check. Picture a number line: 62,030 is to the left of 620,300. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write each side as one number. We can only compare numbers when they are written the usual way.seven hundred thousand = 700,000
700,000 = 700,000 - Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones HTh TTh Th H T O 700,000 7 0 0 0 0 0 700,000 7 0 0 0 0 0 - Compare the digits from the left. The first column where they are different decides which number is bigger.Every column is the same.
- Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.700,000 = 700,000
Check. Both are 700,000. ✓
Watch out. Equal numbers can be written in different ways, so write both as digits first.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Millions Thousands Ones M HTh TTh Th H T O 999,999 9 9 9 9 9 9 1,000,000 1 0 0 0 0 0 0 - Count the digits. A number with more digits is bigger (when no number starts with 0).1,000,000 has 7 digits and 999,999 has 6 digits, so 1,000,000 is bigger.
- Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.999,999 < 1,000,000
Check. Picture a number line: 999,999 is to the left of 1,000,000. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Key words. Ascending order means smallest first. Descending order means biggest first.
Method. Put the numbers in a place-value table, then compare columns from the left. Smallest first.
- Write all the numbers in a place-value table. This lines up the digits so we can compare them.
Thousands Ones HTh TTh Th H T O 534,210 5 3 4 2 1 0 534,120 5 3 4 1 2 0 543,210 5 4 3 2 1 0 53,421 5 3 4 2 1 - Compare the numbers in pairs, from the left. Where the digits in a column match, move on to the next column.53,421 and 534,120: 534,120 has 6 digits and 53,421 has 5 digits, so 534,120 is bigger.
534,120 and 534,210: hundreds: 1 and 2 are different. 1 < 2, so 534,120 is smaller.
534,210 and 543,210: ten thousands: 3 and 4 are different. 3 < 4, so 534,210 is smaller. - Write them smallest first. Check you have used every number once.53,421, 534,120, 534,210, 543,210
Check. Count the numbers: 4 in the question and 4 in the answer. ✓
Watch out. Tick each number off as you write it, so you do not miss one.
Key words. Ascending order means smallest first. Descending order means biggest first.
Method. Put the numbers in a place-value table, then compare columns from the left. Smallest first.
- Write all the numbers in a place-value table. This lines up the digits so we can compare them.
Thousands Ones HTh TTh Th H T O 706,040 7 0 6 0 4 0 760,040 7 6 0 0 4 0 706,400 7 0 6 4 0 0 76,040 7 6 0 4 0 - Compare the numbers in pairs, from the left. Where the digits in a column match, move on to the next column.76,040 and 706,040: 706,040 has 6 digits and 76,040 has 5 digits, so 706,040 is bigger.
706,040 and 706,400: hundreds: 0 and 4 are different. 0 < 4, so 706,040 is smaller.
706,400 and 760,040: ten thousands: 0 and 6 are different. 0 < 6, so 706,400 is smaller. - Write them smallest first. Check you have used every number once.76,040, 706,040, 706,400, 760,040
Check. Count the numbers: 4 in the question and 4 in the answer. ✓
Watch out. Tick each number off as you write it, so you do not miss one.
Key words. Ascending order means smallest first. Descending order means biggest first.
Method. Put the numbers in a place-value table, then compare columns from the left. Smallest first.
- Write all the numbers in a place-value table. This lines up the digits so we can compare them.
Thousands Ones HTh TTh Th H T O 999,001 9 9 9 0 0 1 990,901 9 9 0 9 0 1 909,901 9 0 9 9 0 1 999,010 9 9 9 0 1 0 - Compare the numbers in pairs, from the left. Where the digits in a column match, move on to the next column.909,901 and 990,901: ten thousands: 0 and 9 are different. 0 < 9, so 909,901 is smaller.
990,901 and 999,001: thousands: 0 and 9 are different. 0 < 9, so 990,901 is smaller.
999,001 and 999,010: tens: 0 and 1 are different. 0 < 1, so 999,001 is smaller. - Write them smallest first. Check you have used every number once.909,901, 990,901, 999,001, 999,010
Check. Count the numbers: 4 in the question and 4 in the answer. ✓
Watch out. Tick each number off as you write it, so you do not miss one.
Key words. Ascending order means smallest first. Descending order means biggest first.
Method. Put the numbers in a place-value table, then compare columns from the left. Biggest first.
- Write all the numbers in a place-value table. This lines up the digits so we can compare them.
Thousands Ones HTh TTh Th H T O 250,639 2 5 0 6 3 9 205,639 2 0 5 6 3 9 250,369 2 5 0 3 6 9 25,639 2 5 6 3 9 205,936 2 0 5 9 3 6 - Compare the numbers in pairs, from the left. Where the digits in a column match, move on to the next column.25,639 and 205,639: 205,639 has 6 digits and 25,639 has 5 digits, so 205,639 is bigger.
205,639 and 205,936: hundreds: 6 and 9 are different. 6 < 9, so 205,639 is smaller.
205,936 and 250,369: ten thousands: 0 and 5 are different. 0 < 5, so 205,936 is smaller.
250,369 and 250,639: hundreds: 3 and 6 are different. 3 < 6, so 250,369 is smaller. - Write them biggest first. Check you have used every number once.250,639, 250,369, 205,936, 205,639, 25,639
Check. Count the numbers: 5 in the question and 5 in the answer. ✓
Watch out. Tick each number off as you write it, so you do not miss one.
Key words. Ascending order means smallest first. Descending order means biggest first.
Method. Put the numbers in a place-value table, then compare columns from the left. Biggest first.
- Write all the numbers in a place-value table. This lines up the digits so we can compare them.
Thousands Ones HTh TTh Th H T O 418,000 4 1 8 0 0 0 481,000 4 8 1 0 0 0 4,180 4 1 8 0 408,100 4 0 8 1 0 0 - Compare the numbers in pairs, from the left. Where the digits in a column match, move on to the next column.4,180 and 408,100: 408,100 has 6 digits and 4,180 has 4 digits, so 408,100 is bigger.
408,100 and 418,000: ten thousands: 0 and 1 are different. 0 < 1, so 408,100 is smaller.
418,000 and 481,000: ten thousands: 1 and 8 are different. 1 < 8, so 418,000 is smaller. - Write them biggest first. Check you have used every number once.481,000, 418,000, 408,100, 4,180
Check. Count the numbers: 4 in the question and 4 in the answer. ✓
Watch out. Tick each number off as you write it, so you do not miss one.
Method. Find out what each small mark is worth, then count on from a number you can read.
- Count the gaps between two labelled numbers. The line goes from 400,000 to 500,000 in 10 equal gaps.500,000 − 400,000 = 100,000
- Work out what one gap is worth. Share the distance equally between the gaps.100,000 ÷ 10 = 10,000
- Count on from the start to the arrow. The arrow is 7 gaps after 400,000.400,000 + 7 × 10,000 = 400,000 + 70,000 = 470,000
Check. Count back from 500,000: 3 gaps back is 500,000 − 30,000 = 470,000. ✓
Watch out. Do not read the nearest label. Count the small marks to find the exact number.
Method. Find out what each small mark is worth, then count on from a number you can read.
- Count the gaps between two labelled numbers. The line goes from 50,000 to 60,000 in 10 equal gaps.60,000 − 50,000 = 10,000
- Work out what one gap is worth. Share the distance equally between the gaps.10,000 ÷ 10 = 1,000
- Count on from the start to the arrow. The arrow is 7 gaps after 50,000.50,000 + 7 × 1,000 = 50,000 + 7,000 = 57,000
Check. Count back from 60,000: 3 gaps back is 60,000 − 3,000 = 57,000. ✓
Watch out. Do not read the nearest label. Count the small marks to find the exact number.
Method. Work out what each gap is worth. Then decide which two gaps the arrow is between and how far along it is.
- Work out what one gap is worth. The line goes from 0 to 1,000,000 in 10 equal gaps.1,000,000 ÷ 10 = 100,000
- Find the two numbers either side of the arrow. The arrow is after 2 gaps, but before 3 gaps.200,000 < arrow < 300,000
- Decide how far along the arrow is. Halfway between 200,000 and 300,000 is 250,000.The arrow is close to 250,000.
- Choose the best estimate. Pick the option nearest to the arrow.B (250,000)
Check. The other options are far away: 20,000, 520,000, 720,000. ✓
Watch out. An estimate is a sensible guess. You do not need the exact number.
4?5,200 > 435,200
Method. Try every digit from 0 to 9 in the box and compare each number with the number on the right. Compare from the left.
- Look at the columns that do not change. The hundred thousands digit is 4 in both numbers. The box is in the ten thousands column, so that is the next column to compare.The box digit must be bigger than 3.
- Try each digit. Write each number and compare it.405,200: less than 435,200
415,200: less than 435,200
425,200: less than 435,200
435,200: equal to 435,200
445,200: greater than 435,200 ✓
455,200: greater than 435,200 ✓
465,200: greater than 435,200 ✓
475,200: greater than 435,200 ✓
485,200: greater than 435,200 ✓
495,200: greater than 435,200 ✓ - Write the digits that work. Choose the numbers that are greater.4, 5, 6, 7, 8 and 9
Check. Digit 3 gives exactly 435,200, which is equal, not greater. The next digit is the first to work. ✓
Watch out. > means greater than. An equal number does not count.
Method. The number halfway between two numbers is the same distance from both. Find the gap between them, halve it, then add that to the smaller number.
- Find the gap between the two numbers. Subtract the smaller from the bigger.350,000 − 250,000 = 100,000
- Halve the gap. Halfway is half of the gap away from each number.100,000 ÷ 2 = 50,000
- Add half the gap to the smaller number. This moves us to the middle.250,000 + 50,000 = 300,000
Check. Another way: add the numbers and halve. 250,000 + 350,000 = 600,000 and 600,000 ÷ 2 = 300,000. ✓
Watch out. Halfway is not the average of the digits. Use the whole numbers.
Method. The number halfway between two numbers is the same distance from both. Find the gap between them, halve it, then add that to the smaller number.
- Find the gap between the two numbers. Subtract the smaller from the bigger.175,000 − 125,000 = 50,000
- Halve the gap. Halfway is half of the gap away from each number.50,000 ÷ 2 = 25,000
- Add half the gap to the smaller number. This moves us to the middle.125,000 + 25,000 = 150,000
Check. Another way: add the numbers and halve. 125,000 + 175,000 = 300,000 and 300,000 ÷ 2 = 150,000. ✓
Watch out. Halfway is not the average of the digits. Use the whole numbers.
Method. The number halfway between two numbers is the same distance from both. Find the gap between them, halve it, then add that to the smaller number.
- Find the gap between the two numbers. Subtract the smaller from the bigger.720,000 − 680,000 = 40,000
- Halve the gap. Halfway is half of the gap away from each number.40,000 ÷ 2 = 20,000
- Add half the gap to the smaller number. This moves us to the middle.680,000 + 20,000 = 700,000
Check. Another way: add the numbers and halve. 680,000 + 720,000 = 1,400,000 and 1,400,000 ÷ 2 = 700,000. ✓
Watch out. Halfway is not the average of the digits. Use the whole numbers.
Method. The number halfway between two numbers is the same distance from both. Find the gap between them, halve it, then add that to the smaller number.
- Find the gap between the two numbers. Subtract the smaller from the bigger.99,000 − 1,000 = 98,000
- Halve the gap. Halfway is half of the gap away from each number.98,000 ÷ 2 = 49,000
- Add half the gap to the smaller number. This moves us to the middle.1,000 + 49,000 = 50,000
Check. Another way: add the numbers and halve. 1,000 + 99,000 = 100,000 and 100,000 ÷ 2 = 50,000. ✓
Watch out. Halfway is not the average of the digits. Use the whole numbers.
Method. Think about which digit goes in the hundred thousands column, then the ten thousands, and so on. The biggest columns matter most.
- Put the digits in order, biggest first. The biggest digit in the biggest column makes the biggest number.9, 7, 5, 3, 2, 0
- Write them in that order. Biggest digit in the hundred thousands column, the next in the ten thousands, and so on.975,320
Check. Try swapping two digits. The new number is not bigger than 975,320. ✓
Watch out. A six-digit number cannot start with a zero.
Method. Think about which digit goes in the hundred thousands column, then the ten thousands, and so on. The biggest columns matter most.
- Put the digits in order, smallest first. The smallest digit in the biggest column makes the smallest number.0, 2, 3, 5, 7, 9
- Check the first digit. A six-digit number cannot start with 0, because then it would only have five digits. Use the smallest digit that is not 0 first, then put the 0 straight after it.First digit 2
- Write the number. 203,579
Check. Try swapping two digits. The new number is not smaller than 203,579. ✓
Watch out. A six-digit number cannot start with a zero.
Method. Think about which digit goes in the hundred thousands column, then the ten thousands, and so on. The biggest columns matter most.
- A number is even if its last digit (the ones digit) is even. Even digits are 0, 2, 4, 6, 8.Even digits here: 0, 2
- Try each possible ones digit. Make the rest of the number as small as possible each time. The other five digits go smallest first (but not starting with 0).Ones digit 0: smallest number is 235,790
Ones digit 2: smallest number is 305,792 - Choose the smallest of these numbers. Compare from the left.The smallest even number is 235,790.
Check. Try swapping two digits. The new number is not smaller than 235,790. ✓
Watch out. A six-digit number cannot start with a zero.
Method. Think about which digit goes in the hundred thousands column, then the ten thousands, and so on. The biggest columns matter most.
- Which digits can go first? The number must be less than 500,000, so the first digit must be smaller than 5 (and not 0).Possible first digits: 1, 2, 4
- Choose the biggest first digit. The first digit matters most.First digit = 4
- Put the other digits in order, biggest first. This makes the number as close to the limit as possible.4 then 8, 6, 2, 1, 0
- Write the number. 486,210
Check. Try swapping two digits. The new number is not bigger than 486,210. ✓
Watch out. A six-digit number cannot start with a zero.
Method. Think about which digit goes in the hundred thousands column, then the ten thousands, and so on. The biggest columns matter most.
- Which digits can go first? The number must be more than 500,000, so the first digit must be bigger than 5 (and not 0).Possible first digits: 6, 8
- Choose the smallest first digit. The first digit matters most.First digit = 6
- Put the other digits in order, smallest first. This makes the number as close to the limit as possible.6 then 0, 1, 2, 4, 8
- Write the number. 601,248
Check. Try swapping two digits. The new number is not smaller than 601,248. ✓
Watch out. A six-digit number cannot start with a zero.
| Town | People |
|---|---|
| Ashby | 185,420 |
| Bexley | 158,240 |
| Corton | 185,240 |
| Dunmore | 98,765 |
| Elford | 205,010 |
Method. Put the numbers in order and pick the one you need. Compare the numbers from the left.
- Write all the numbers in a place-value table. This lines them up.
Thousands Ones HTh TTh Th H T O 185,420 1 8 5 4 2 0 158,240 1 5 8 2 4 0 185,240 1 8 5 2 4 0 98,765 9 8 7 6 5 205,010 2 0 5 0 1 0 - Put them in order, biggest first. Compare column by column from the left.1. Elford: 205,010
2. Ashby: 185,420
3. Corton: 185,240
4. Bexley: 158,240
5. Dunmore: 98,765 - Pick the one with the most people. It is number 1 in the list.Elford (205,010)
Check. All the other numbers are smaller than 205,010 in the way we need. ✓
Watch out. Read the question again: is it the biggest, the smallest or the second biggest?
Method. Put the numbers in order and pick the one you need. Compare the numbers from the left.
- Write all the numbers in a place-value table. This lines them up.
Thousands Ones HTh TTh Th H T O 185,420 1 8 5 4 2 0 158,240 1 5 8 2 4 0 185,240 1 8 5 2 4 0 98,765 9 8 7 6 5 205,010 2 0 5 0 1 0 - Put them in order, smallest first. Compare column by column from the left.1. Dunmore: 98,765
2. Bexley: 158,240
3. Corton: 185,240
4. Ashby: 185,420
5. Elford: 205,010 - Pick the one with the fewest people. It is number 1 in the list.Dunmore (98,765)
Check. All the other numbers are different than 98,765 in the way we need. ✓
Watch out. Read the question again: is it the biggest, the smallest or the second biggest?
Method. Put the populations in descending order, then write the town names in that order.
- Put the populations in a place-value table. This lines up the digits.
Thousands Ones HTh TTh Th H T O 185,420 1 8 5 4 2 0 158,240 1 5 8 2 4 0 185,240 1 8 5 2 4 0 98,765 9 8 7 6 5 205,010 2 0 5 0 1 0 - Put them in order, biggest first. Compare from the left. 185,420 and 185,240 are the same until the hundreds column: 4 is more than 2.205,010 → Elford
185,420 → Ashby
185,240 → Corton
158,240 → Bexley
98,765 → Dunmore - Write the town names in that order. Elford, Ashby, Corton, Bexley, Dunmore
Check. There are 5 towns in the table and 5 in the answer. ✓
Watch out. Be careful with 185,420 and 185,240: they use the same digits but in different columns.
| Stadium | Seats |
|---|---|
| Arena A | 62,400 |
| Arena B | 59,250 |
| Arena C | 74,500 |
| Arena D | 60,000 |
Method. Put the numbers in order and pick the one you need. Compare the numbers from the left.
- Write all the numbers in a place-value table. This lines them up.
Thousands Ones TTh Th H T O 62,400 6 2 4 0 0 59,250 5 9 2 5 0 74,500 7 4 5 0 0 60,000 6 0 0 0 0 - Put them in order, biggest first. Compare column by column from the left.1. Arena C: 74,500
2. Arena A: 62,400
3. Arena D: 60,000
4. Arena B: 59,250 - Pick the one with the second most seats. It is number 2 in the list.Arena A (62,400)
Check. All the other numbers are different than 62,400 in the way we need. ✓
Watch out. Read the question again: is it the biggest, the smallest or the second biggest?
Method. Compare the two numbers properly, starting from the left. Then say what went wrong with Mei’s reasoning.
- Count the digits. 98,765 has 5 digits. 100,234 has 6 digits.A number with more digits is bigger.
- Compare the numbers. The six-digit number is bigger.98,765 < 100,234
- Say what Mei did wrong. She compared only the first digits (9 and 1). But the 1 in 100,234 is in the hundred thousands column and is worth 100,000, while the 9 in 98,765 is in the ten thousands column and is worth only 90,000.The first digit is worth different amounts because the numbers have different lengths.
Check. 100,234 − 98,765 = 1,469, so 100,234 is 1,469 more. ✓
Watch out. Always count the digits before you compare the first digits.
“If you swap two digits in a number, the new number is smaller.”
Method. Try swapping digits in a few numbers. If you find one that gets bigger and one that gets smaller, the answer is “sometimes”.
- Swap the first two digits of 123,456. The 1 and 2 swap.123,456 → 213,456. Is 213,456 bigger? Yes. It got bigger.
- Swap the first two digits of 654,321. The 6 and 5 swap.654,321 → 564,321. Is 564,321 smaller? Yes. It got smaller.
- Swap two digits that are the same in 440,000. The two 4s swap.440,000 stays 440,000. It did not change.
- Write the answer. It can get bigger, smaller or stay the same.Sometimes (it is not always smaller)
Check. Moving a bigger digit to a bigger column makes the number bigger. ✓
Watch out. One example is not enough to say “always”. Try to find an example that breaks the rule.
Method. Work out each multiplication, or divide the big number by the small number.
- Check Amira. Multiply 60,000 by 10.60,000 × 10 = 600,000 ✓
- Check Noah. Multiply 6,000 by 100.6,000 × 100 = 600,000 ✓
- Say who is right. Both statements work out to 600,000.Both Amira and Noah are right.
Check. On a place-value chart, the 6 moves one place left to go from 60,000 to 600,000 (10 times) and two places left from 6,000 (100 times). ✓
Watch out. Each place to the left makes a number 10 times bigger.
Method. The number must start with 4 or 5 to be close to 500,000. Find the closest number above and the closest below, then see which is nearer.
- Closest number above 500,000. It starts with 5. Put the other digits in order, smallest first, to stay as close to 500,000 as possible.5, then 1, 2, 3, 4, 6 → 512,346
512,346 − 500,000 = 12,346 - Closest number below 500,000. It starts with 4. Put the other digits in order, biggest first.4, then 6, 5, 3, 2, 1 → 465,321
500,000 − 465,321 = 34,679 - Compare the two gaps. The smaller gap is the closer number.12,346 < 34,679, so 512,346 is closer.
Check. Any other number is further away. For example 513,246 is 13,246 away. ✓
Watch out. Check numbers on both sides of 500,000. The closest number is not always above or always below.