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Primary Maths · Year 4Chapter 1 · Place value to 10,000
Year 4 · Chapter 1

Year 4 Place Value: worked examples and questions with answers

By the end of this chapter you will be able to

  • Count on and back in steps of 6, 7, 9, 25 and 1000
  • Say the value of every digit in a four-digit number and write numbers in digits and words
  • Find 10, 100 or 1000 more or less than a number
  • Compare and order numbers beyond 1000 and estimate where they sit on a number line
  • Round numbers to the nearest 10, 100 or 1000
  • Count backwards through zero and use negative numbers, for example for temperatures
  • Read and write Roman numerals from I to C and explain why zero and place value make our number system better

You should already know

  • Place value of three-digit numbers (hundreds, tens and ones)
  • Counting in 2s, 3s, 4s, 5s, 8s, 10s, 50s and 100s
  • Comparing and ordering numbers up to 1000
  • Roman numerals to XII

Where this leads

  • Adding and subtracting four-digit numbers (Chapter 2)
  • Multiplying and dividing by 10 and 100 (Chapter 4)
  • Decimals and money (Chapter 6)
  • Negative numbers and place value to 1,000,000 in Year 5

National Curriculum for England, Key Stage 2 mathematics, Year 4. Content: Count in multiples of 6, 7, 9, 25 and 1000, Find 1000 more or less than a number, Count backwards through zero to include negative numbers, Know the place value of each digit in a four-digit number, Order and compare numbers beyond 1000, Identify, represent and estimate numbers in different ways, Round any number to the nearest 10, 100 or 1000, Solve number and practical problems with larger positive numbers, Read Roman numerals to 100 (I to C) and know how the number system changed to include zero and place value.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FWhat is the value of the 6 in 365?
2
FWhat is 100 more than 456?
3
FWrite these numbers in order, smallest first:   407,   470,   74,   704
4
FCount on in 50s. Write the next four numbers after 350.
5
FWrite 9 in Roman numerals.
1.1

Counting in steps of 6, 7, 9, 25 and 1000

Year 4
Count in multiples

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Counting in steps of 6, 7, 9, 25 and 1000Coming soon

Counting in steps means adding the same number again and again. When you start at 0 and count in 6s, you say 6, 12, 18, 24, 30… These numbers are the multiples of 6. They are the answers in the 6 times table. Here is a hundred square with the multiples of 6 coloured in. Can you see the pattern?

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100
The multiples of 6 on a hundred square. They make a zig-zag.

Counting in 7s. When a jump crosses a tens number, jump to the tens number first. For 28 + 7: 28 + 2 = 30, then 30 + 5 = 35.

07142128354249+7+7+7+7+7+7+7
Counting in 7s: every jump is +7.

Counting in 9s. Adding 9 is the same as adding 10 and then taking away 1. So 36 + 9 is 36 + 10 = 46, then 46 − 1 = 45. The multiples of 9 go 9, 18, 27, 36, 45… The tens digit goes up by 1 and the ones digit goes down by 1.

Counting in 25s. Think of 25p coins. Four 25p coins make 100p, which is £1. So the last two digits go 25, 50, 75, 00, and then they start again.

0255075100125150175200+25+25+25+25+25+25+25+25
Counting in 25s: 25, 50, 75, 100, 125, 150, 175, 200.

Counting in 1000s. Only the thousands digit changes. Counting on from 2,350 we say 3,350, 4,350, 5,350… Counting back from 2,350 we say 1,350 and then 350. You can count back in every step too: take away the step each time.

Key words. A step is the jump you make each time. A multiple of a number is a number you get when you count in steps of that number from 0.

Key ideas

  • Counting on means adding the step each time. Counting back means taking it away.
  • Multiples of 9: add 10 and take away 1.
  • Multiples of 25 end in 25, 50, 75 or 00.
  • Counting in 1000s: only the thousands digit changes.
  • A number is a multiple of 7 if you say it when you count in 7s from 0.
Worked example 1.1.1
Count on in 6s from 18. Write the next four numbers.

Key words. Counting in steps of 6 means adding 6 again and again. The numbers you say are multiples of 6 if you start from 0.

Method. Counting on in 6s means adding 6 each time. Use each new number to start the next jump.

  1. Find the size of each jump. We are counting on in steps of 6.
    Each jump is +6.
  2. Work out the numbers one jump at a time. Always start from the number you have just found.
    18 + 2 = 20 (a tens number), then + 4 more = 24
    24 + 6 = 30
    30 + 6 = 36
    36 + 4 = 40 (a tens number), then + 2 more = 42
  3. Write the numbers in order.
    24, 30, 36, 42

Check. Go the other way from the last number: 42 − 6 = 36. ✓

Watch out. Take care when you cross a tens number. Jump to the next ten first, then add the rest.

Worked example 1.1.2
Count back in 9s from 90. Write the next four numbers.

Key words. Counting in steps of 9 means taking away 9 again and again. The numbers you say are multiples of 9 if you start from 0.

Method. Counting back in 9s means taking away 9 each time. Use each new number to start the next jump.

  1. Find the size of each jump. We are counting back in steps of 9.
    Each jump is −9.
  2. Work out the numbers one jump at a time. Always start from the number you have just found.
    90 − 10 = 80, then add 1 back = 81
    81 − 10 = 71, then add 1 back = 72
    72 − 10 = 62, then add 1 back = 63
    63 − 10 = 53, then add 1 back = 54
  3. Write the numbers in order.
    81, 72, 63, 54

Check. Go the other way from the last number: 54 + 9 = 63. ✓

Watch out. Adding 9 is the same as adding 10 and then taking away 1. Taking away 9 is taking away 10 and adding 1 back.

Worked example 1.1.3
Write the missing numbers in this sequence.   4,350,  ?,  ?,  7,350

Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.

  1. Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.
    From 4,350 to 7,350 is 3 jumps.
    The difference is 3,000.
    3,000 ÷ 3 = 1,000
  2. Say which way and how big each jump is. The numbers are getting bigger, so we count on.
    Each jump is +1,000.
  3. Fill in the missing numbers. Add (or take away) one jump from a number you know.
    Number 2: 4,350 + 1,000 = 5,350
    Number 3: 7,350 − 1,000 = 6,350
  4. Write the missing numbers.
    5,350, 6,350

Check. Read the whole sequence again and check every jump is +1,000: 4,350, 5,350, 6,350, 7,350. ✓

Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.

Worked example 1.1.4
Write the missing numbers in this sequence.   175,  ?,  225,  250,  ?

Method. Find the size of one jump from two numbers you can see. Then use it to fill the gaps.

  1. Find two numbers you know and count the jumps between them. Each jump in the sequence is the same size.
    From 175 to 225 is 2 jumps.
    The difference is 50.
    50 ÷ 2 = 25
  2. Say which way and how big each jump is. The numbers are getting bigger, so we count on.
    Each jump is +25.
  3. Fill in the missing numbers. Add (or take away) one jump from a number you know.
    Number 2: 175 + 25 = 200
    Number 5: 250 + 25 = 275
  4. Write the missing numbers.
    200, 275

Check. Read the whole sequence again and check every jump is +25: 175, 200, 225, 250, 275. ✓

Watch out. Do not just look at the first two numbers. Check the jump is the same all the way along.

Worked example 1.1.5
Is 52 a multiple of 6? Count in 6s to find out.

Method. A multiple of 6 is a number you say when you count in 6s. Count in 6s until you reach or pass 52.

  1. Count in 6s. Stop when you get to 52 or go past it.
    6, 12, 18, 24, 30, 36, 42, 48, 54
  2. Did we land exactly on 52? If we land on it, it is a multiple. If we jump over it, it is not.
    We reached 54. We jumped from 48 to 54 and missed 52, so no.
  3. Write the answer.
    Is 52 a multiple of 6? No

Check. 6 × 8 = 48 and 6 × 9 = 54. 52 is in between. ✓

Watch out

The first number you say is the first jump, not the number you start on. If you count on in 6s from 18, the first number you say is 24, not 18. Also take extra care when a jump crosses a tens or hundreds number, like 98 + 7.

Exercise 1A  Counting in steps

Non-calculator
1
FCount on in 6s. Write the next three numbers each time. The hundred square can help you.
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100
(a)Start at 12
(b)Start at 30
(c)Start at 48
(d)Start at 54
2
FCount on in 7s. Write the next three numbers each time.
(a)Start at 14
(b)Start at 35
(c)Start at 49
(d)Start at 63
3
FCount on in 9s. Write the next three numbers each time.
(a)Start at 18
(b)Start at 45
(c)Start at 72
(d)Start at 81
4
FCount back. Write the next three numbers each time.
(a)In 6s from 60
(b)In 7s from 70
(c)In 9s from 99
(d)In 25s from 300
5
FZainab is counting in 25s. Write the numbers that are missing from the number line.
0???100???200
6
FCount in 1000s. Write the next four numbers each time.
(a)Count on from 2,350
(b)Count back from 9,125
(c)Count on from 5,008
(d)Count back from 10,000
7
FWrite the missing numbers in each sequence.
(a)7,  14,  ?,  28,  35,  ?
(b)81,  ?,  63,  ?,  45
(c)?,  375,  400,  ?,  450
(d)3,600,  ?,  ?,  6,600,  ?
(e)60,  ?,  48,  ?,  36
8
FQuick-fire! Is the first number a multiple of the second? Write yes or no.
(a)42  &  6
(b)52  &  7
(c)81  &  9
(d)77  &  7
(e)175  &  25
(f)7,000  &  1,000
9
PSAmira puts a 25p coin in her money jar every day. How many days will it take her to save £2?
10
PSA box holds 6 eggs. Count in 6s to find how many eggs there are in 9 boxes.
11
PSA library has 4,250 books. Every month 1,000 new books arrive and no books leave. How many books will the library have after 5 months?
12
RJack counts in 7s:   7, 14, 21, 27, 35, 42.   He has made a mistake. Spot the mistake and write the correct number.
13
RWhich of these numbers is not in the 9 times table?   36,   45,   56,   63
14
RZainab counts in 9s from 0. Leo counts in 6s from 0. What is the first number above 0 that they both say?
1.2

Thousands, hundreds, tens and ones

Year 4
Place value, Represent numbers

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Thousands, hundreds, tens and onesComing soon

A four-digit number has four columns: thousands (Th), hundreds (H), tens (T) and ones (O). Each column is worth 10 times the column to its right. 10 ones make 1 ten. 10 tens make 1 hundred. 10 hundreds make 1 thousand.

You can show a number with base-ten blocks. A big cube is 1,000, a flat is 100, a rod is 10 and a small cube is 1. These blocks show 2,314.

2,314
Base-ten blocks for 2,314: 2 thousands, 3 hundreds, 1 ten and 4 ones.

You can also use place-value counters. Each counter in a column has the value of that column. Here is 4,306. The tens column has no counters, so the digit there is a 0. The zero is a place holder. It keeps the other digits in the right columns. Without it, 4,306 would look like 436.

Th10001000100010004H1001001003T0O1111116
Place-value counters for 4,306.

Partitioning means splitting a number into the value of each digit: 4,306 = 4,000 + 300 + 6. We write numbers in words too: 4,306 is four thousand, three hundred and six. We put “and” before the last two digits.

Key words. The value of a digit is what it is worth. A digit is one of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A place holder is a zero that holds an empty column.

Key ideas

  • Columns, right to left: ones, tens, hundreds, thousands. Each is 10 times bigger than the one before.
  • The value of a digit is the digit times the value of its column: the 7 in 3,742 is worth 700.
  • A zero is a place holder: 4,306 and 436 are very different numbers.
  • Partition: 4,306 = 4,000 + 300 + 6.
  • Words: put “and” before the last two digits.
Worked example 1.2.1
What is the value of the underlined digit in 5,247?

Key words. The value of a digit is how much it is worth. A place-value table has columns: ones (O), tens (T), hundreds (H), thousands (Th).

Method. The value of a digit depends on its column. Find the column, then work out the digit times the column value.

  1. Write the number in a place-value table. Each column has its own name and value.
    ThHTO
    5247
  2. Find which column the digit is in. The coloured digit is 2, in the hundreds column.
    1 hundred = 100
  3. Multiply the digit by the value of the column. There are 2 hundreds.
    2 × 100 = 200

Check. Say it in words: 2 hundreds is two hundred. ✓

Watch out. The digit is 2, but its value is 200. The value depends on where the digit sits.

Worked example 1.2.2
Partition 6,305. Write it as an addition.

Key words. To partition a number is to split it into parts, one part for each digit.

Method. Partition means splitting a number into the value of each of its digits.

  1. Write the number in a place-value table. This shows how many thousands, hundreds, tens and ones there are.
    ThHTO
    6305
  2. Work out the value of each digit. Digit times the value of its column.
    6 thousands = 6,000
    3 hundreds = 300
    0 tens = nothing to write
    5 ones = 5
  3. Leave out the zeros. A zero means that column is empty, so there is nothing to add.
  4. Write the values as an addition. Start with the biggest value.
    6,305 = 6,000 + 300 + 5

Check. Add them up: 6,000 + 300 + 5 = 6,305. ✓

Watch out. Write 3,000 for 3 thousands, not just 3. Each part keeps its full value.

Worked example 1.2.3
What number do these blocks show?

Method. Count each kind of block, work out what each kind is worth, then add them up.

  1. Count the big cubes. A big cube is worth 1,000.
    1 big cube = 1,000
  2. Count the flats. A flat is worth 100.
    4 flats = 400
  3. Count the rods and the small cubes. A rod is worth 10. A small cube is worth 1.
    2 rods = 20
    3 small cubes = 3
  4. Add the values. Start with the biggest.
    1,000 + 400 + 20 + 3 = 1,423

Check. Say it in words: one thousand, four hundred and twenty-three. ✓

Watch out. A flat is 100, not 10. Count each kind of block separately.

Worked example 1.2.4
Write 7,030 in words.

Method. Say the number in parts, from the biggest column to the smallest: thousands, then hundreds, then the last two digits.

  1. Write the number in a place-value table. This helps us see each part.
    ThHTO
    7030
  2. Say each part in words. Put "thousand" after the thousands and "hundred" after the hundreds.
    7 thousands → seven thousand
    30 (the last two digits) → thirty
  3. Join the parts together. Put "and" before the last part, just before the tens and ones.
    seven thousand and thirty

Check. Read it out loud. It sounds like 7,030. ✓

Watch out. The words for 4,065 are four thousand and sixty-five. There are no hundreds, so you do not say "hundred".

Worked example 1.2.5
Write “four thousand and sixty-five” in digits.

Method. Listen for the words "thousand" and "hundred". They tell you which column each digit goes in.

  1. Split the words into parts. Each part belongs to one or two columns.
    four thousand → 4 in the thousands column
    no hundreds are said → 0 in the hundreds column
    sixty-five → 6 tens and 5 ones
  2. Write the digits in the place-value table. Write 0 in any column that was not mentioned.
    ThHTO
    4065
  3. Write the number.
    4,065

Check. Say 4,065 in words: four thousand and sixty-five. ✓

Watch out. If no tens or ones are said, those columns are zeros: “six thousand” is 6,000.

Watch out

Do not forget the zero place holder. “Four thousand and sixty-five” is 4,065, not 465 and not 4,650. Also remember that the value of a digit is not the digit itself: in 6,420 the 6 is worth 6,000.

Exercise 1B  Place value in four-digit numbers

Non-calculator
1
FWhat number do these base-ten blocks show?
2
FWhat number do these place-value counters show?
Th100010001000H100100TO1111
3
FWhat is the value of the underlined digit?
(a)4,752
(b)6,039
(c)8,914
(d)2,356
4
FPartition each number. Write it as an addition.
(a)3,452
(b)7,086
(c)5,300
(d)9,019
5
FWrite each number in digits.
(a)5,000 + 300 + 60 + 2
(b)8,000 + 40 + 9
(c)2,000 + 700
(d)6 thousands, 0 hundreds, 8 tens and 1 one
6
FWrite each number in digits.
(a)four thousand, five hundred and twelve
(b)six thousand and forty
(c)eight thousand, three hundred
(d)two thousand and nine
7
FWrite each number in words.
(a)2,365
(b)5,009
(c)7,800
(d)3,070
8
PSHere are four digit cards. Use all four cards each time to make a four-digit number.
6039
(a)The largest number you can make
(b)The smallest number you can make
(c)The smallest even number you can make
9
FHow many are there?   Use what you know about place value.
(a)How many tens are in 560?
(b)How many hundreds are in 3,400?
(c)How many hundreds are in 7,300?
10
PSKofi has 3 boxes of 1,000 screws, 7 boxes of 100 screws, 2 boxes of 10 screws and 5 loose screws. How many screws does he have?
11
RMei says, “The 6 in 6,420 has a value of 6.” Is she right? Explain.
12
RRavi writes “four thousand and sixty-five” as 4,650. Spot his mistake and write the correct number.
13
RWhich one is different?   5,030    5,000 + 30    5 thousands and 3 tens    5,300
14
RI am a four-digit number.
My hundreds digit is 0.
My thousands digit is twice my tens digit.
My ones digit is 5 more than my tens digit.
My digits add up to 13.
What number am I?
1.3

1000 more or less, comparing and ordering

Year 4
More and less, Order and compare

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: 1000 more or less, comparing and orderingComing soon

10, 100 or 1000 more or less. Find the right column, then change that digit by 1. For 1000 more than 4,572, add 1 to the thousands digit: 4,572 becomes 5,572. Take care with 9s and 0s. 1000 more than 9,450 is 10,450, because 9 thousands + 1 thousand = 10 thousands. 100 less than 3,050 is 2,950, because there are no hundreds, so we exchange 1 thousand for 10 hundreds.

Comparing. Put the numbers in a place-value table and compare from the left. The first column that is different decides.

ThHTO
4,5624562
4,6524652
The thousands match (4 and 4). The hundreds are different: 5 is less than 6. So 4,562 < 4,652.

We use two signs: < means “is less than” and > means “is greater than”. The wide mouth opens towards the bigger number. To order numbers, compare them in pairs. Ascending means smallest first. Descending means biggest first.

Numbers on a number line. A number line can show numbers beyond 1000. First work out what each small mark is worth, then count on. You can also estimate: if an arrow points about halfway between 7,000 and 8,000, a good estimate is 7,500.

4,0004,5005,000?
The line goes from 4,000 to 5,000 in 10 steps of 100. The arrow points to 4,300.

Key ideas

  • To find 1000 more or less, change the thousands digit by 1. For 10 or 100, change the tens or hundreds digit.
  • Watch out for 9s (carry) and 0s (exchange).
  • Compare numbers from the left. The first column that is different decides.
  • < means less than. > means greater than.
  • A number with more digits is bigger (when no number starts with 0).
Worked example 1.3.1
What is 1000 more than 4,572?

Method. To find 1,000 more, change the thousands digit by 1. Look out for 9s (carrying) and 0s (exchanging).

  1. Find the thousands digit. 1,000 is one thousand, so only the thousands digit (and maybe the next one) should change.
    ThHTO
    4572
  2. Add 1 to the thousands digit. 1,000 more means 1 more thousand.
    4 + 1 = 5 thousands
  3. Write the new number. Digits in the other columns stay the same unless we carried or exchanged.
    4,572 + 1,000 = 5,572

Check. Do the opposite: 5,572 − 1,000 = 4,572. ✓

Watch out. Do not change every digit! Only the digit in the column you are working on changes (unless you carry or exchange).

Worked example 1.3.2
What is 100 less than 3,050?

Method. To find 100 less, change the hundreds digit by 1. Look out for 9s (carrying) and 0s (exchanging).

  1. Find the hundreds digit. 100 is one hundred, so only the hundreds digit (and maybe the next one) should change.
    ThHTO
    3050
  2. Take 1 from the hundreds digit. There are 0 hundreds, so we must exchange 1 thousand for 10 hundreds. Then take 1 away.
    0 hundreds → take 1 from the thousands digit, then 10 − 1 = 9 hundreds
  3. Write the new number. Digits in the other columns stay the same unless we carried or exchanged.
    3,050 − 100 = 2,950

Check. Do the opposite: 2,950 + 100 = 3,050. ✓

Watch out. Do not change every digit! Only the digit in the column you are working on changes (unless you carry or exchange).

Worked example 1.3.3
What is 1000 more than 9,450?

Method. To find 1,000 more, change the thousands digit by 1. Look out for 9s (carrying) and 0s (exchanging).

  1. Find the thousands digit. 1,000 is one thousand, so only the thousands digit (and maybe the next one) should change.
    ThHTO
    9450
  2. Add 1 to the thousands digit. There are 9 thousands. 1 more thousand makes 10 thousands, which is 1 ten thousand and 0 thousands. So we carry.
    9 + 1 = 10 → write 0 in the thousands column and carry 1 to the ten thousands column
  3. Write the new number. Digits in the other columns stay the same unless we carried or exchanged.
    9,450 + 1,000 = 10,450

Check. Do the opposite: 10,450 − 1,000 = 9,450. ✓

Watch out. Do not change every digit! Only the digit in the column you are working on changes (unless you carry or exchange).

Worked example 1.3.4
Write < or > in the box:   4,572  [ ? ]  4,527

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.

  1. Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
    ThHTO
    4,5724572
    4,5274527
  2. Compare the digits from the left. The first column where they are different decides which number is bigger.
    thousands: 4 and 4 are the same, so look at the next column.
    hundreds: 5 and 5 are the same, so look at the next column.
    tens: 7 and 2 are different. 7 > 2, so 4,572 is bigger.
  3. Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.
    4,572 > 4,527

Check. Picture a number line: 4,527 is to the left of 4,572. ✓

Watch out. Start at the LEFT (the biggest column). Do not start with the ones.

Worked example 1.3.5
Write these in order, smallest first:   6,104,   6,014,   6,410,   1,604,   6,041

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.

  1. Write all the numbers in a place-value table. This lines up the digits so we can compare them.
    ThHTO
    6,1046104
    6,0146014
    6,4106410
    1,6041604
    6,0416041
  2. Compare the thousands, then the hundreds, then the tens, then the ones. Where the digits in a column match, move on to the next column.
    1,604 and 6,014: thousands: 1 and 6 are different. 1 < 6, so 1,604 is smaller.
    6,014 and 6,041: tens: 1 and 4 are different. 1 < 4, so 6,014 is smaller.
    6,041 and 6,104: hundreds: 0 and 1 are different. 0 < 1, so 6,041 is smaller.
    6,104 and 6,410: hundreds: 1 and 4 are different. 1 < 4, so 6,104 is smaller.
  3. Write them smallest first. Check you have used every number once.
    1,604, 6,014, 6,041, 6,104, 6,410

Check. Count the numbers: 5 in the question and 5 in the answer. ✓

Watch out. Do not forget a number. Tick each one off as you write it.

Worked example 1.3.6
What number is the arrow pointing to?
4,0004,5005,000?

Method. Find out what each small mark is worth, then count on from a number you can read.

  1. Count the gaps between two labelled numbers. The line goes from 4,000 to 5,000 in 10 equal gaps.
    5,000 − 4,000 = 1,000
  2. Work out what one gap is worth. Share the distance equally between the gaps.
    1,000 ÷ 10 = 100
  3. Count on from the start to the arrow. The arrow is 7 gaps after 4,000.
    4,000 + 7 × 100 = 4,000 + 700 = 4,700

Check. Count back from 5,000: 3 gaps back is 5,000 − 300 = 4,700. ✓

Watch out. Do not read the nearest label. Count the small marks to find the exact number.

Watch out

When you find 1000 more or less, do not change the other digits. 1000 less than 3,208 is 2,208 (only the thousands digit changes), not 3,198. And when you compare, start on the left: 4,562 is smaller than 4,652 even though the ones digit 2 is the same.

Exercise 1C  More, less, compare and order

Non-calculator
1
FWrite the number that is 1000 more than each number.
(a)2,345
(b)6,908
(c)5,090
(d)8,999
2
FWrite the number that is 1000 less than each number.
(a)6,432
(b)9,075
(c)4,008
(d)7,000
3
FWork out each answer.
(a)10 more than 3,495
(b)100 more than 4,950
(c)10 less than 6,005
(d)100 less than 7,040
(e)100 more than 9,900
4
FWrite the missing number in each box.
(a)? + 1,000 = 7,368
(b)5,120 − ? = 4,120
(c)? − 100 = 3,950
(d)2,406 + ? = 3,406
5
FWrite <, > or = in each box.
(a)4,562  [ ? ]  4,652
(b)7,009  [ ? ]  7,090
(c)3,200  [ ? ]  3,000 + 200
(d)8,451  [ ? ]  845
(e)6,730  [ ? ]  6,073
(f)5,000 + 40  [ ? ]  5,400
6
FWrite each set of numbers in order, smallest first.
(a)4,385,  4,358,  4,835,  4,538
(b)6,014,  6,104,  6,041,  6,410,  1,604
(c)2,999,  3,001,  2,990,  3,010
7
FWrite each set of numbers in order, biggest first.
(a)7,280,  7,028,  7,208,  782
(b)9,099,  9,909,  9,090,  9,990,  9,009
8
FWhat number is the arrow pointing to?
7,0007,5008,000?
9
PSThe arrow on the number line points to a number between 7,000 and 8,000. Which is the best estimate?   A  7,100    B  7,500    C  7,900    D  8,500
05,0007,0008,00010,000?
10
FAnswer these questions about four-digit numbers.
(a)What is the smallest four-digit number?
(b)What is the largest four-digit number?
(c)What is 1 more than the largest four-digit number?
11
PSThe town of Elmby has 6,450 people. Brookley has 1000 more people than Elmby. Oakton has 100 fewer people than Elmby.
(a)How many people live in Brookley?
(b)How many people live in Oakton?
(c)Which of the three towns has the fewest people?
12
RLeo says, “1000 less than 3,208 is 3,198.” Leo is wrong. What mistake did he make? What is the correct answer?
13
RIs this always, sometimes or never true? Give an example.
“The number with the bigger digit total is the bigger number.”
14
RUse each of the digits 1, 4, 6 and 9 once to make a four-digit number. Write all the even numbers that are greater than 6,000.
1.4

Rounding to the nearest 10, 100 or 1000

Year 4
Rounding, Estimate

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Rounding to the nearest 10, 100 or 1000Coming soon

Sometimes we do not need an exact number. A newspaper might say “about 3,500 people came to the match” when the real number was 3,467. Changing a number to a nearby, simpler number is called rounding. We round to the nearest 10, 100 or 1000.

Look at 3,467 on this number line. It is between 3,460 and 3,470. The halfway point is 3,465. 3,467 is past halfway, so it is nearer to 3,470.

3,4603,4653,470halfway3,467
3,467 to the nearest 10: past halfway, so round up to 3,470.

Here is the rule. Look at the digit to the right of the place you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down. The digits after the rounding place become zeros.

For 3,467 to the nearest 100: look at the tens digit, 6. It is 5 or more, so round up to 3,500. For 3,467 to the nearest 1000: look at the hundreds digit, 4. It is 4 or less, so round down to 3,000. When a number is exactly halfway, such as 6,500, we always round up.

3,4003,4503,500halfway3,467
3,467 to the nearest 100: between 3,400 and 3,500. Past halfway (3,450), so 3,500.
3,0003,5004,000halfway3,467
3,467 to the nearest 1000: between 3,000 and 4,000. Before halfway (3,500), so 3,000.

Rounding helps us estimate. To estimate 4,968 + 3,012, round to the nearest 1000: 5,000 + 3,000 = 8,000. The real answer is close to 8,000.

Key words. To round is to change a number to a nearby, simpler one. An estimate is a sensible guess that is close to the exact answer. ≈ means “is about equal to”.

Key ideas

  • Look at the digit to the right of the place you are rounding to.
  • 5 or more: round up. 4 or less: round down.
  • Digits to the right of the rounding place become zeros.
  • Halfway always rounds up: 6,500 to the nearest 1000 is 7,000.
  • A 9 can carry: 9,960 to the nearest 100 is 10,000.
Worked example 1.4.1
Round 3,467 to the nearest 10.

Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.

Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.

  1. Find the two multiples of 10 that 3,467 is between. Rounding to the nearest 10 means choosing the closest multiple of 10.
    3,460  <  3,467  <  3,470
  2. Picture it on a number line. The halfway point between 3,460 and 3,470 is 3,465.
    3,4603,4653,470halfway3,467
  3. Look at the ones digit (the digit just to the right of the tens digit). This digit tells us if we are past halfway.
    The ones digit is 7.
  4. Decide: round up or round down? 5 or more: round up. 4 or less: round down.
    7 is 5 or more, so round up.
  5. Write the rounded number. Digits to the right of the tens column become zeros.
    3,467 ≈ 3,470 (to the nearest 10)

Check. Distances: 3,467 − 3,460 = 7 and 3,470 − 3,467 = 3. 3,470 is the nearer one. ✓

Watch out. Only the digit just to the right decides. Do not look at digits further along.

Worked example 1.4.2
Round 5,382 to the nearest 100.

Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.

Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.

  1. Find the two multiples of 100 that 5,382 is between. Rounding to the nearest 100 means choosing the closest multiple of 100.
    5,300  <  5,382  <  5,400
  2. Look at the tens digit (the digit just to the right of the hundreds digit). This digit tells us if we are past halfway.
    The tens digit is 8.
  3. Decide: round up or round down? 5 or more: round up. 4 or less: round down.
    8 is 5 or more, so round up.
  4. Write the rounded number. Digits to the right of the hundreds column become zeros.
    5,382 ≈ 5,400 (to the nearest 100)

Check. Distances: 5,382 − 5,300 = 82 and 5,400 − 5,382 = 18. 5,400 is the nearer one. ✓

Watch out. Only the digit just to the right decides. Do not look at digits further along.

Worked example 1.4.3
Round 6,500 to the nearest 1000.

Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.

Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.

  1. Find the two multiples of 1,000 that 6,500 is between. Rounding to the nearest 1,000 means choosing the closest multiple of 1,000.
    6,000  <  6,500  <  7,000
  2. Look at the hundreds digit (the digit just to the right of the thousands digit). This digit tells us if we are past halfway.
    The hundreds digit is 5.
  3. Decide: round up or round down? 5 or more: round up. 4 or less: round down.
    5 is 5 or more, so round up.
  4. Write the rounded number. Digits to the right of the thousands column become zeros.
    6,500 ≈ 7,000 (to the nearest 1,000)

Check. Distances: 6,500 − 6,000 = 500 and 7,000 − 6,500 = 500. 7,000 is the one we choose when exactly halfway. ✓

Watch out. Only look at the digit just to the right. Do not round twice. Halfway (like 6,500 to the nearest 1,000) always rounds up.

Worked example 1.4.4
Round 7,962 to the nearest 100.

Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.

Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.

  1. Find the two multiples of 100 that 7,962 is between. Rounding to the nearest 100 means choosing the closest multiple of 100.
    7,900  <  7,962  <  8,000
  2. Look at the tens digit (the digit just to the right of the hundreds digit). This digit tells us if we are past halfway.
    The tens digit is 6.
  3. Decide: round up or round down? 5 or more: round up. 4 or less: round down.
    6 is 5 or more, so round up.
  4. Write the rounded number. Digits to the right of the hundreds column become zeros.
    7,962 ≈ 8,000 (to the nearest 100)

Check. Distances: 7,962 − 7,900 = 62 and 8,000 − 7,962 = 38. 8,000 is the nearer one. ✓

Watch out. Only the digit just to the right decides. Do not look at digits further along.

Watch out

Only look at the digit just to the right. To round 3,459 to the nearest 10, the ones digit 9 tells us to round up to 3,460. Do not round the 9 up to make the 5 into a 6 and then round again. Also remember: 5 rounds up, so 3,465 to the nearest 10 is 3,470.

Exercise 1D  Rounding and estimating

Non-calculator
1
FRound each number to the nearest 10.
(a)63
(b)78
(c)145
(d)2,384
(e)5,495
(f)7,001
2
FRound each number to the nearest 100.
(a)340
(b)3,650
(c)6,238
(d)4,950
(e)8,449
(f)9,960
3
FRound each number to the nearest 1000.
(a)4,280
(b)6,500
(c)7,999
(d)2,499
(e)9,501
(f)1,350
4
FThe number line shows 4,372. Use it to round 4,372 to the nearest 100.
4,3004,3504,400halfway4,372
5
FThe arrow points to a number. Round that number to the nearest 1000.
6,0006,5007,000?
6
FRound 5,764 to the nearest:
(a)10
(b)100
(c)1000
7
PSWhich digits could go in the box so that 4,3?6 rounds to 4,400 when rounded to the nearest 100?
8
PSA newspaper rounds crowd sizes to the nearest 1000. On Saturday 5,610 people came to a match. On Sunday 5,390 people came to the match.
(a)What number does the newspaper write for Saturday?
(b)What number does the newspaper write for Sunday?
(c)The newspaper says “The crowd was the same on both days.” Is that true? Explain.
9
PSEstimate the answer by rounding each number first.
(a)4,968 + 3,012  (round to the nearest 1,000)
(b)7,098 − 2,947  (round to the nearest 1,000)
(c)6,214 + 2,788  (round to the nearest 100)
10
RWhich of these numbers does not round to 5,000 when rounded to the nearest 1000?   4,501   5,499   5,500   4,999
11
RAisha says, “3,456 rounded to the nearest 10 is 3,500.” Spot her mistake. What is the correct answer?
12
RTom says, “The smallest whole number that rounds to 700 (to the nearest 100) is 650, and the largest is 749.” Is Tom right? Explain.
13
PSRuby’s school is 3,845 metres from her home.
(a)Round this distance to the nearest 100 metres.
(b)Round this distance to the nearest 1000 metres. How many kilometres is that?
14
RUse each of the digits 7, 5, 3 and 2 once to make a four-digit number that rounds to 7,000 when rounded to the nearest 1000. Write all the numbers that work.
1.5

Negative numbers: counting back through zero

Year 4
Negative numbers

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Negative numbers: counting back through zeroComing soon

It can be so cold that the temperature goes below zero. A temperature of three degrees below zero is written −3°C. Numbers below zero are called negative numbers. They have a minus sign in front. Look at the thermometer: the liquid is below the 0 line.

−10−505101520
A thermometer showing −4°C. The numbered marks go up in 5s. Each small mark is 1 degree.

On a number line, zero is in the middle. Numbers to the right are positive and get bigger. Numbers to the left are negative. The further left a number is, the smaller it is. So −9 is smaller than −2.

−10−9−8−7−6−5−4−3−2−1012345678910−43
On a number line, numbers get bigger as you go right. −4 is less than 3.

Counting back through zero. If you count back in 3s from 9, you say 6, 3, 0, −3, −6… After 0 the numbers keep going down, but they are negative. When a jump crosses zero, split it. To work out 2 − 5, go down 2 to reach 0, then 3 more to reach −3.

The gap between two temperatures. To find how many degrees from −4°C to 5°C, count up to 0 (4 degrees) and then up to 5 (5 more degrees): 4 + 5 = 9 degrees.

Key words. A negative number is a number less than zero. Degrees Celsius (°C) measure temperature. Zero is neither positive nor negative.

Key ideas

  • Negative numbers are less than zero and have a minus sign: −1, −2, −3…
  • On a number line, the further left, the smaller the number.
  • Counting back through zero: … 2, 1, 0, −1, −2 …
  • To cross zero, split the jump into the part that reaches 0 and the part that goes past it.
  • The gap between a negative and a positive number: add the two distances from zero.
Worked example 1.5.1
Count back in 3s from 12. Write the next six numbers.

Key words. A negative number is less than zero. It has a minus sign in front, like −4. Think of temperatures below freezing.

Method. Counting back in 3s means taking away 3 each time. A number line helps when we go through zero.

  1. Draw a number line. Numbers to the left of 0 are negative (below zero). Numbers to the right are positive.
    −7−6−5−4−3−2−1012345678910111213
  2. Jump back 3 each time, starting from 12. When we pass zero the numbers become negative.
    12 − 3 = 9
    9 − 3 = 6
    6 − 3 = 3
    3 − 3 = 0
    0 − 3: go down 0 to reach 0, then 3 more below zero → −3
    −3 − 3: start below zero and go down 3 more → −6
  3. Write the numbers in order.
    9, 6, 3, 0, −3, −6

Check. Go the other way from the last number: −6 + 3 = −3. ✓

Watch out. After 0 comes −1, not 1. The numbers below zero count down: −1, −2, −3...

Worked example 1.5.2
The temperature is 4°C. It falls by 7°C. What is the new temperature?

Method. Draw a number line, start at 4 and jump down 7. When you pass 0 the numbers change between positive and negative.

  1. Show the start on a number line. This lets us see where zero is.
    −5−4−3−2−101234564−7
  2. Go down 7. Split the jump into two parts: the part that reaches 0, then the part that goes past it.
    4 − 7: go down 4 to reach 0, then 3 more below zero → −3
  3. Write the answer.
    −3°C

Check. Do the opposite: −3 + 7 = 4. ✓

Watch out. Going down past zero gives a negative number. Going up from a negative number makes it closer to zero.

Worked example 1.5.3
Write these numbers in order, smallest first:   −5,   3,   −8,   0,   2

Method. On a number line, numbers further to the left are smaller. Mark each number, then read them off from left to right.

  1. Put the numbers on a number line. The line shows which are below zero.
    −9−8−7−6−5−4−3−2−101234
  2. Read from left to right. Left is smallest, right is biggest.
    −8, −5, 0, 2, 3
  3. Write them smallest first.
    −8, −5, 0, 2, 3

Check. The numbers below zero are the smallest. Of those, the one with the biggest digit (like −9) is the furthest from zero, so it is the smallest. ✓

Watch out. −9 is smaller than −2. The further below zero, the smaller the number.

Worked example 1.5.4
How many degrees warmer is 5°C than −4°C?

Method. The difference between two numbers is how far apart they are on a number line.

  1. Mark both numbers on a number line. Seeing the gap helps.
    −6−5−4−3−2−101234567−459
  2. Count the gap. One number is below zero and one is above. Count up to zero first, then on to the other number.
    −4 up to 0 is 4.
    0 up to 5 is 5.
    4 + 5 = 9
  3. Write the answer.
    9°C

Check. Start at −4 and go up 9: −4 + 9 = 5. ✓

Watch out. Do not just subtract the digits when one number is negative. Go through zero: −4 and 5 are 9 apart, not 1.

Worked example 1.5.5
What temperature does this thermometer show?
−10−505101520

Method. Find the numbered mark nearest the top of the red liquid, then count the small marks to the top.

  1. Look at the numbers on the scale. The scale is numbered every 5 degrees. The small marks are 1 degree each.
    Big numbers: ..., −5, 0, 5, 10, ...
  2. Find the numbered mark just below the top of the red. The top of the red is between −5 and 0.
    The mark below is −5.
  3. Count the small marks above it. Each small mark is 1 degree.
    2 small marks above −5: −5 + 2 = −3

Check. Count from 0 instead: the top of the red is 3 small marks below 0, so the temperature is −3°C. ✓

Watch out. Going up the scale the numbers get bigger. Below zero, the number nearest the bottom is the smallest.

Watch out

The number with the bigger digit is not always the bigger number. −9 is smaller than −2, because it is further below zero. And after 0 comes −1, not 1. A number line helps you check.

Exercise 1E  Negative numbers

Non-calculator
1
FWhat temperature does the thermometer show?
−10−505101520
2
FWrite the missing numbers in the boxes on the number line.
−10?−5?0?5?10
3
FCount on or back. Write the next numbers each time.
(a)Count back in 2s from 7. Write 5 numbers.
(b)Count back in 5s from 20. Write 5 numbers.
(c)Count on in 3s from −10. Write 5 numbers.
(d)Count back in 10s from 40. Write 6 numbers.
4
FWrite < or > in each box.
(a)−3  [ ? ]  2
(b)−7  [ ? ]  −4
(c)0  [ ? ]  −1
(d)−10  [ ? ]  −12
(e)−5  [ ? ]  5
(f)−8  [ ? ]  −18
5
FWrite each set of numbers in order.
(a)3,  −5,  0,  −8,  2  (smallest first)
(b)−12,  −2,  7,  −7,  12  (smallest first)
(c)−4,  −14,  4,  −41,  14  (biggest first)
6
FWhat is the new temperature?
(a)It is 5°C. It falls by 8°C.
(b)It is −2°C. It rises by 6°C.
(c)It is −4°C. It falls by 3°C.
(d)It is −9°C. It rises by 9°C.
7
PSThe thermometers show the temperatures in three cities one morning.
−15−10−50510Oslo−15−10−50510Leeds−15−10−50510Moscow
(a)Which city is the coldest?
(b)How many degrees warmer is Leeds than Oslo?
(c)How many degrees colder is Moscow than Oslo?
8
FFind the difference between the two temperatures.
(a)−2°C and 7°C
(b)−8°C and −3°C
(c)−6°C and 4°C
(d)5°C and −9°C
9
PSIn a shop the floors below the ground floor are negative numbers (floor −1 is the first basement). The lift is on floor 2. It goes down 5 floors. Which floor is the lift on now?
10
PSA diver is 12 metres below the surface of the sea, which is at −12 m. She swims up 5 m and then dives down 9 m. What is her position now?
11
RPriya says, “−9 is bigger than −2 because 9 is bigger than 2.” Is she right? Explain.
12
RIs this always, sometimes or never true?
“A two-digit negative number is smaller than a one-digit negative number.”
13
PSWhich of these temperatures are colder than −6°C?   3°C,   −9°C,   −2°C,   −6°C,   −10°C
14
RAt midnight the temperature is −4°C. It rises by 3°C every hour.
(a)What is the temperature after 2 hours?
(b)After how many hours is the temperature first above 0°C?
(c)After 3 hours the temperature falls by 5°C. What is the temperature then?
1.6

Roman numerals and the story of zero

Year 4
Roman numerals, Zero and place value

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Roman numerals and the story of zeroComing soon

The Romans used letters for numbers. You can still see Roman numerals on clocks, buildings and book chapters. We need five symbols to read every number from 1 to 100:

I1V5X10L50C100
The five Roman symbols you need: I = 1, V = 5, X = 10, L = 50, C = 100.

The rules. (1) Symbols that are the same add up: XX = 10 + 10 = 20. (2) A smaller symbol after a bigger one is added: XI = 10 + 1 = 11. (3) A smaller symbol before a bigger one is taken away: IX = 10 − 1 = 9. (4) You never write the same symbol more than three times in a row, so 4 is IV (not IIII) and 40 is XL (not XXXX). To write a number, do the tens first, then the ones.

OnesRomanTensRoman
1I10X
2II20XX
3III30XXX
4IV40XL
5V50L
6VI60LX
7VII70LXX
8VIII80LXXX
9IX90XC
100C

Why do we not use Roman numerals today? The Romans had no symbol for zero, and the position of a symbol did not matter: X always means 10, wherever it is. That makes big calculations very hard. Try adding XLVII and LXXIX without changing them into our numbers!

Our number system came about over many hundreds of years. Mathematicians in India, and then in the Middle East and Europe, developed the ten digits 0 to 9. The digit 0 became a number in its own right and also a place holder. The key idea is place value: a digit is worth more or less depending on its column. In 22, the first 2 means 20 and the second 2 means 2. In XX, both Xs mean 10. Place value and zero make it much easier to write and calculate with big numbers.

Key words. A numeral is a symbol for a number. Place value means the value of a digit depends on where it is. A place holder is a zero that keeps the other digits in the right columns.

Key ideas

  • I = 1, V = 5, X = 10, L = 50, C = 100.
  • Same symbols add up. A smaller symbol after adds. A smaller symbol before takes away (IV = 4, IX = 9, XL = 40, XC = 90).
  • Never more than three of the same symbol in a row.
  • Write the tens first, then the ones: 74 = LXX + IV = LXXIV.
  • Roman numerals have no zero and no place value. Our number system has both.
Worked example 1.6.1
Write 47 in Roman numerals.

Key words. Roman numerals are the letters the Romans used for numbers: I = 1, V = 5, X = 10, L = 50, C = 100.

Method. Split the number into tens and ones, write each part in Roman numerals, then put them side by side (tens first).

  1. Split the number into tens and ones. Roman numerals are built from the tens first and then the ones.
    47 = 40 + 7
  2. Write the tens in Roman numerals. Use X for 10, L for 50, C for 100. For 40 we write XL (10 before 50) and for 90 we write XC (10 before 100).
    40 = XL
  3. Write the ones in Roman numerals. Use I for 1, V for 5. For 4 we write IV (1 before 5) and for 9 we write IX (1 before 10).
    7 = VII
  4. Put the tens first, then the ones. The tens part goes on the left, the ones part on the right.
    XL + VII = XLVII

Check. Read it back: XL (50 − 10) + V (5) + I (1) + I (1) = 47. ✓

Watch out. Never write the same symbol more than three times in a row. 4 is IV, not IIII, and 40 is XL, not XXXX.

Worked example 1.6.2
What number is LXXIX?

Key words. A numeral is a symbol for a number. Roman numerals use the letters I, V, X, L and C.

Method. Split the Roman numeral into groups, write the value of each group, then add them up.

  1. Split the numeral into groups, reading from the left. A small symbol in front of a bigger one (like IV or XL) is taken away. Otherwise just add.
    L = 50
    X = 10
    X = 10
    IX = 10 − 1 = 9   (small number in front, so take it away)
  2. Add the values. Add the bigger groups first, then the small ones.
    50 + 10 + 10 + 9 = 79
  3. Write the answer.
    79

Check. Write 79 in Roman numerals: tens first, then ones: LXX + IX = LXXIX. ✓

Watch out. Read from the left. A small symbol only gets taken away when it comes BEFORE a bigger one: IX is 9 but XI is 11.

Worked example 1.6.3
Write 94 in Roman numerals.

Key words. Roman numerals are the letters the Romans used for numbers: I = 1, V = 5, X = 10, L = 50, C = 100.

Method. Split the number into tens and ones, write each part in Roman numerals, then put them side by side (tens first).

  1. Split the number into tens and ones. Roman numerals are built from the tens first and then the ones.
    94 = 90 + 4
  2. Write the tens in Roman numerals. Use X for 10, L for 50, C for 100. For 40 we write XL (10 before 50) and for 90 we write XC (10 before 100).
    90 = XC
  3. Write the ones in Roman numerals. Use I for 1, V for 5. For 4 we write IV (1 before 5) and for 9 we write IX (1 before 10).
    4 = IV
  4. Put the tens first, then the ones. The tens part goes on the left, the ones part on the right.
    XC + IV = XCIV

Check. Read it back: XC (100 − 10) + IV (5 − 1) = 94. ✓

Watch out. Never write the same symbol more than three times in a row. 4 is IV, not IIII, and 40 is XL, not XXXX.

Worked example 1.6.4
Work out XVIII + XXIV. Give your answer in Roman numerals.

Method. Change the Roman numerals into numbers, do the calculation, then change the answer back into Roman numerals.

  1. Change XVIII into a number. Split it into groups and add.
    XVIII = X (10) + V (5) + I (1) + I (1) + I (1) = 18
  2. Change XXIV into a number. Do the same.
    XXIV = X (10) + X (10) + IV (5 − 1) = 24
  3. Do the calculation. Use our normal numbers.
    18 + 24 = 42
  4. Change the answer back to Roman numerals. Tens first, then ones.
    42 = XL + II = XLII (42)

Check. Read XLII again: 40 + 1 + 1 = 42. ✓

Watch out. Roman numerals are hard to add directly. Change them to numbers first.

Watch out

Do not write 8 as IIX or 49 as IL. A small symbol goes in front of a bigger one to take away only in a few cases: I can go before V or X, and X can go before L or C. 49 is XL (40) + IX (9) = XLIX, not IL. And remember: the same symbol is never written more than three times in a row.

Exercise 1F  Roman numerals

Non-calculator
1
FWrite each Roman numeral as a number. The chart shows the symbols.
I1V5X10L50C100
(a)VII
(b)XIV
(c)XXIX
(d)XLI
(e)LXVIII
(f)XCIX
2
FWrite each number in Roman numerals.
(a)9
(b)14
(c)23
(d)38
(e)44
(f)76
3
FQuick-fire! Write each number in Roman numerals.
(a)30
(b)50
(c)60
(d)90
(e)100
(f)40
4
FWrite < or > between each pair of Roman numerals.
(a)XLIX  [ ? ]  LI
(b)LXIV  [ ? ]  LXVI
(c)XC  [ ? ]  LXXXIX
5
FWrite these Roman numerals in order, smallest first.   XXXIV,   XLIII,   XXIV,   XL,   LX
6
FWork out each answer. Write the answer in Roman numerals, with the number in brackets.
(a)IX + XI
(b)XV + XXV
(c)LXII + XVII
(d)C − XXV
(e)L − XIV
(f)XC − XLII
7
PSThe clock face uses Roman numerals. What time does the clock show? Write the time using numbers.
XIIIIIIIIIVVVIVIIVIIIIXXXI
8
RNoah writes 8 as IIX and 49 as IL. Both are wrong. Write the correct Roman numerals.
9
RExplain why we need the digit 0 to write the number three thousand and forty (3,040). What would happen without it?
10
RIn the number 22, the two 2s do not mean the same thing. In the Roman numeral XX, the two Xs do mean the same thing. Explain.
11
RWrite true or false for each sentence.
(a)The Romans had a symbol for zero in their numerals.
(b)In our number system the value of a digit depends on its column (place value).
(c)The digits 0 to 9 that we use today were developed over many hundreds of years, starting in India.
12
PSBooks on a shelf are numbered with Roman numerals. The first book is XL and the last book is LV. How many books are on the shelf?
13
RWhich number from 1 to 100 needs the most letters when it is written in Roman numerals? How many letters does it need?

Chapter summary

SkillMethod
Counting in stepsAdd (or take away) the same number each time. For 9: +10 then −1. For 25: 25, 50, 75, 00. For 1000: change the thousands digit.
Place valueTh, H, T, O. Value = digit × column value. Zero is a place holder. 4,306 = 4,000 + 300 + 6.
1000 more or lessChange the thousands digit by 1. Watch for carrying (9) and exchanging (0).
Comparing and orderingCompare from the left. The first column that is different decides. < less than, > greater than.
Estimating on a number lineWork out what each small mark is worth, then count on. Halfway between 7,000 and 8,000 is 7,500.
RoundingLook at the digit to the right. 5 or more: round up. 4 or less: round down. Halfway rounds up.
Negative numbersLeft of zero on a number line. The further left, the smaller. Split jumps that cross zero.
Roman numeralsI = 1, V = 5, X = 10, L = 50, C = 100. Tens first, then ones. A small symbol before a bigger one is taken away.
Zero and place valueOur system has a zero and place value, so a digit’s value depends on its column. Roman numerals have neither.

Can you…?

  • I can count on and back in 6s, 7s, 9s, 25s and 1000s.
  • I can say the value of every digit in a four-digit number and write numbers in words.
  • I can find 10, 100 or 1000 more or less than a number.
  • I can compare and order numbers beyond 1000 and use < and >.
  • I can estimate numbers on a number line.
  • I can round to the nearest 10, 100 or 1000.
  • I can count backwards through zero and compare negative numbers.
  • I can read and write Roman numerals up to 100.
  • I can explain why zero and place value are important.

Key terms

Multiple
A number you say when you count in steps of that number from 0. The multiples of 6 are 6, 12, 18…
Digit
One of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Place value
The value of a digit depends on its column: ones, tens, hundreds or thousands.
Place holder
A zero that holds an empty column so the other digits stay in the right place, as in 4,065.
Partition
To split a number into the value of each digit: 4,306 = 4,000 + 300 + 6.
Ascending
Getting bigger: smallest first.
Descending
Getting smaller: biggest first.
Round
To change a number to a nearby, simpler number, such as to the nearest 10, 100 or 1000.
Estimate
A sensible guess that is close to the exact answer.
Negative number
A number less than zero, written with a minus sign, like −5.
Roman numerals
Letters used for numbers by the Romans: I, V, X, L, C.
Numeral
A symbol that stands for a number.

Exercise 1Z  Check-up (non-calculator)

Non-calculator
1
Count on in 7s. Write the next three numbers after 28.
2
What is the value of the underlined digit in 3,847?
3
Write 6,050 in words.
4
What is 1000 less than 5,206?
5
Write < or > in the box:   6,089  [ ? ]  6,098
6
Round 3,578 to the nearest 100.
7
The temperature is 2°C. It falls by 6°C. What is the new temperature?
8
Write in order, smallest first:   −6,   2,   −1,   −10
9
Write 64 in Roman numerals.
10
Tom counts on in 25s from 350. He says four numbers. What is the last number he says?

Total 17 marks. Allow about 17 minutes.

Exercise 1X  SATs-style questions

Non-calculator
1
FWrite the number that is 1000 more than 6,745.[1 mark]
2
FWhich number has 5 tens?   Tick one box.   3,502   5,032   3,052   3,520[1 mark]
3
FWrite seven thousand and forty in digits.[1 mark]
4
FRound 5,962 to the nearest 100.[1 mark]
5
FWrite the missing numbers in the sequence.
4,350,  5,350,  ?,  ?,  8,350[2 marks]
6
RAisha rounds 4,450 to the nearest 1000. She says the answer is 5,000. Is Aisha correct? Explain how you know.[2 marks]
7
PSAt 6 a.m. the temperature was −5°C. By noon it had risen by 9°C. What was the temperature at noon?[2 marks]
8
FWrite these numbers in order, smallest first.   4,096   4,960   4,069   4,906[2 marks]
9
PSIn a quiz Layla scores LXXIV points and Ruby scores XCI points. How many more points does Ruby score than Layla? Write your answer as a number.[2 marks]
10
RMei thinks of a four-digit number.
When she rounds it to the nearest 100 she gets 5,300.
When she rounds it to the nearest 1000 she gets 5,000.
The tens digit is 3 and the ones digit is 9.
What is Mei’s number? Show how you know.[3 marks]
Chapter 2: Adding and subtracting →

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