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Primary Maths · Year 6Chapter 1 · Place value to 10,000,000 and negative numbers
Year 6 · Chapter 1

Year 6 Place Value and Negative Numbers: worked examples and questions with answers

By the end of this chapter you will be able to

  • Read and write numbers up to 10,000,000 in figures and words, and say the value of every digit
  • Compare and order numbers up to 10,000,000 and use the signs < and >
  • Make the largest and smallest numbers from digit cards
  • Round any whole number to the nearest 10, 100, 1,000, 10,000, 100,000 or 1,000,000, and explain the halfway rule
  • Order negative numbers and count forwards and backwards through zero
  • Work out intervals across zero, using steps to zero from each number
  • Solve number and practical problems that use place value, rounding and negative numbers

You should already know

  • Place value of numbers up to 1,000,000 (Year 5, Chapter 1)
  • Rounding to the nearest 10, 100, 1,000, 10,000 and 100,000 (Year 5, Chapter 1)
  • Counting back through zero, for example with temperatures (Year 5, Chapter 1)

Where this leads

  • Multiplying and dividing whole numbers by 10, 100 and 1,000, and using written methods with large numbers (Chapter 2)
  • Place value of decimals to three decimal places (Chapter 5)
  • Rounding to a degree of accuracy in estimates and in the SATs revision chapter (Chapter 13)

National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Read, write, order and compare numbers up to 10,000,000 and determine the value of each digit, Round any whole number to a required degree of accuracy, Use negative numbers in context, and calculate intervals across zero, Solve number and practical problems that involve all of the above.

Exercise P  Do you remember?

Non-calculator
1
FWhat is the value of the 7 in 4,732?
MillionsThousandsOnes
TMMHThTThThHTO
4732
2
FRound 4,736 to the nearest 1,000.
4,0005,0004,500halfway4,736
3
FWhat is 1,000 more than 25,400?
4
FCount back from 3 to −2. Write every number you say.
5
FHow many hundreds are there in 5,000?
1.1

Reading, writing and the value of each digit

Year 6
Read, write and order numbers up to 10,000,000, Determine the value of each digit in a number
Video lesson: Reading, writing and the value of each digitComing soon

Numbers up to 10,000,000 can have up to eight digits. We split them into groups of three, counting from the right. The first group is the ones, the second group is the thousands, and the third group is the millions. A comma goes between the groups, so 4,672,819 is easy to read.

Every digit has a value that depends on its column. Moving one column to the left makes the value ten times bigger. In a place-value table the columns are: ten millions (TM), millions (M), hundred thousands (HTh), ten thousands (TTh), thousands (Th), hundreds (H), tens (T) and ones (O).

MillionsThousandsOnes
TMMHThTThThHTO
4672819
The number 4,672,819 in a place-value table. The shaded 6 is in the hundred thousands column, so it is worth 600,000.

To read a big number, read each group and say its name. For 3,050,400 you say “three million, fifty thousand, four hundred”. Zeros are not said, but they hold the columns in place. A number with a zero in the middle can be surprising: 4,000,080 is four million and eighty.

Ten million is written 10,000,000. It has eight digits and seven zeros. One million, 1,000,000, has seven digits and six zeros. Each time you add a column, the number is ten times bigger.

Key ideas

  • Put a comma after every group of three digits, counting from the right.
  • The value of a digit is the digit times the value of its column: 6 in the hundred thousands column is 6 × 100,000 = 600,000.
  • Each column is ten times the column on its right, so 10 thousands make 1 ten thousand.
  • 1,000,000 is one million. It is also 1,000 thousands and 10 hundred thousands.
  • Zeros in the middle still count: 4,000,080 is four million and eighty, because the zeros hold the columns in place.
  • Read the number in groups: millions, thousands, and then the ones group.
Worked example 1.1.1
What is the value of the 6 in 4,672,819?

Key words. The value of a digit depends on its column (its place). Each column is ten times the column on its right. In 4,672,819 the 6 is in the hundred thousands column, so it is worth 6 × 100,000 = 600,000.

Method. Find the column that holds the digit, then multiply the digit by the value of that column.

  1. Find the column of the 6. Count the columns from the right. The 6 is in the hundred thousands column.
    4,672,819: the 6 is in the hundred thousands column
  2. Multiply the digit by the column value. The value of a digit is the digit times the value of its column.
    6 × 100,000 = 600,000
  3. Check the column. Count the columns from the right to be sure.
    Column 6 from the right is the hundred thousands column

Check. Only the digit 6 is worth 600,000. The other digits are worth their own values, so the whole number stays 4,672,819. ✓

Watch out. The 6 is worth 600,000, not 6. Its column decides its value.

Worked example 1.1.2
Write 3,050,400 in words.

Method. Split the number into groups of three digits, then read each group and say its name.

  1. Split the number into groups. Each group of three has its own name: million, thousand, or the ones group.
    3,050,400
  2. Read each group in words. Say the number in each group, then its name.
    three million | fifty thousand | four hundred
  3. Join the groups with commas. Use "and" before a last group that is less than 100.
    three million, fifty thousand, four hundred

Check. Write the words back as figures: 3,050,400. ✓

Watch out. Zeros still count. 4,000,080 is four million and eighty, not four thousand and eighty.

Worked example 1.1.3
Write this number in figures: six million, two hundred and nine thousand and seventy.

Method. Find the value of each group, then add the groups together.

  1. Find the value of each group. Each name tells you how many zeros the group needs.
    6,000,000 + 209,000 + 70
  2. Add the groups. Put them together to make one number.
    6,000,000 + 209,000 + 70 = 6,209,070
  3. Check the groups. Read the number back in words to check.
    six million | two hundred and nine thousand | seventy

Check. Reading 6,209,070 back gives "six million, two hundred and nine thousand and seventy". ✓

Watch out. Put commas after every three digits from the right: 6,209,070.

Worked example 1.1.4
Write 7,406,512 as the sum of the value of each digit.

Method. Write each digit with its column value, and leave out the zeros.

  1. Find the digit in each column. Use the place-value columns, from the left.
    7 4 0 6 5 1 2
  2. Multiply each digit by its column value. A digit is worth its digit times its column value.
    7,000,000 + 400,000 + 6,000 + 500 + 10 + 2
  3. Add the parts. Adding the parts gives the number back.
    7,000,000 + 400,000 + 6,000 + 500 + 10 + 2 = 7,406,512

Check. 7,000,000 + 400,000 + 6,000 + 500 + 10 + 2 gives 7,406,512 again. ✓

Watch out. Do not write 7 + 4 + 6 + 5 + 1 + 2. Each digit has its own value.

Watch out

Do not read a number by the first digit only. In 4,000,080 the 8 is worth 80, not 8, and the zeros between the 4 and the 8 are needed. Also, the 7 in 1,702,000 is worth 700,000, not 7,000: the 7 is in the hundred thousands column.

Exercise 1A  Reading, writing and place value

Non-calculator
1
FWrite the value of the 5 in 5,284,190.
MillionsThousandsOnes
TMMHThTThThHTO
5284190
2
FWrite the value of each digit that is asked for.
(a)The digit in the hundred thousands column of 1,423,600
(b)The digit in the ten thousands column of 2,093,541
(c)The digit in the thousands column of 807,650
(d)The digit in the tens column of 4,061,580
3
FWrite each number in words.
(a)3,050,400
(b)1,206,070
(c)9,000,009
4
FWrite each number in figures.
(a)four million, sixty thousand and twelve
(b)two million, nine hundred and fifty
(c)seven million, one hundred thousand
5
FWrite 7,406,512 as the sum of the value of each digit.
6
PSJamal says the 7 in 1,702,000 is worth 7,000. Is he right? Explain your answer.
7
FHow many thousands are there in 2 million?
8
FHow many ten thousands are there in 1 million?
9
FWrite the number that has 6 millions, 4 ten thousands, 7 hundreds and 9 ones. All other columns are zero.
10
FWhich digit is in the ten thousands column of 2,358,914?
11
FWhat is the value of the 1 in 8,910,000?
MillionsThousandsOnes
TMMHThTThThHTO
8910000
12
RRuby says, “Ten million has six zeros.” Is she right? Explain your answer.
1.2

Comparing and ordering numbers

Year 6
Compare and order numbers up to 10,000,000, Use the signs < and >, Make the largest and smallest numbers from digit cards
Video lesson: Comparing and ordering numbersComing soon

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.

MillionsThousandsOnes
NumberTMMHThTThThHTO
4,628,1004628100
4,618,9004618900
Compare 4,628,100 and 4,618,900. Both have the same millions, and the same hundred thousands. The ten thousands digits are 2 and 1, so the first number is bigger.

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.
Worked example 1.2.1
Which is bigger, 4,628,100 or 4,618,900? Use the sign to show your answer.

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.

  1. 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.
  2. 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
  3. 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.

Worked example 1.2.2
Put these numbers in order from the smallest: 3,040,000; 3,400,000; 304,000; 3,000,450.

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.

  1. 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
  2. Write the numbers from the smallest. Put the smallest number first.
    304,000, 3,000,450, 3,040,000, 3,400,000
  3. 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.

Worked example 1.2.3
Use the digits 9, 0, 5, 3, 7, 1 and 2, each once, to make the largest 7-digit number.

Method. Put the biggest digit in the leftmost column, then the next biggest, and so on.

  1. Sort the digits from biggest to smallest. The leftmost column has the biggest value.
    9 7 5 3 2 1 0
  2. Write the digits in that order. Each digit goes in the next column to the right.
    9,753,210
  3. 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
1
FWrite the correct sign, < or >, between each pair of numbers.
MillionsThousandsOnes
NumberTMMHThTThThHTO
6,250,0006250000
6,205,0006205000
(a)6,250,000  ?  6,205,000
(b)3,409,000  ?  3,490,000
(c)1,008,000  ?  1,080,000
2
FWrite the correct sign, < or >, between each pair of numbers.
(a)950,000  ?  1,000,000
(b)4,305,000  ?  435,000
(c)2,999,999  ?  3,000,000
3
FWrite these numbers in order, from the smallest to the largest: 2,409,000;  240,900;  2,490,000;  2,040,900;  2,900,400.
4
FWrite these numbers in order, from the largest to the smallest: 5,000,001;  5,100,000;  4,999,999;  5,010,000.
5
FUse the digits 4, 8, 0, 2, 9, 1 and 6, each once, to make the largest 7-digit number.
6
FUse the same seven digits, each once, to make the smallest 7-digit number.
7
PSWhich of these numbers is closest to 5,000,000?   A  4,820,000   B  5,060,000   C  4,990,000   D  5,400,000
8
PSWhich numbers from this list are between 3,000,000 and 4,000,000?   3,090,000; 4,010,000; 2,999,999; 3,999,999
9
FWrite the number that is halfway between 4,000,000 and 5,000,000.
10
FWhat is 1 more than the largest 6-digit number?
11
RAlways, sometimes or never? A number with seven digits is bigger than a number with six digits. Explain your answer.
12
RWrite the number that is 1 more than the smallest 7-digit number that has no zeros in it.
1.3

Rounding to a degree of accuracy

Year 6
Round any whole number to a required degree of accuracy, Use rounding to estimate, Use the halfway point to decide
Video lesson: Rounding to a degree of accuracyComing soon

We round a number when an exact answer is not needed, or when a number is too long to use easily. The degree of accuracy tells you the place value to round to: the nearest 10, 100, 1,000, 10,000, 100,000 or 1,000,000.

To round, find the two multiples of that place value either side of the number. Look at the digit just to the right of the rounding digit. If it is 5 or more, the number is nearer to the bigger multiple, so round up. If it is less than 5, round down. Every digit after the rounding digit then becomes a zero in the answer for that place.

When the number is exactly halfway, for example 350,000 to the nearest 100,000, we round up.

The same method works for negative whole numbers. Ignore the minus sign, round the number, then put the minus sign back. So −6,548 rounds to −7,000 to the nearest 1,000, because 6,548 rounds to 7,000.

6,540,0006,550,0006,545,000halfway6,548,213
Rounding 6,548,213 to the nearest 10,000. The number is past the halfway point of 6,545,000, so it rounds up to 6,550,000.

Key ideas

  • To round, find the two multiples of the place value on each side of the number.
  • Look at the digit to the right of the rounding digit. 5 or more means round up. 4 or less means round down.
  • The digits after the rounding place become zeros: 6,548,213 to the nearest 10,000 is 6,550,000.
  • Halfway (for example 500,000 to the nearest 1,000,000) rounds up.
  • For a negative number, ignore the minus sign, round, then put the minus sign back.
  • Rounding is for estimates and checks. Use the exact number when a question asks for it.
Worked example 1.3.1
Round 6,548,213 to the nearest 10,000.

Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.

Method. Find the two multiples of 10,000 either side of the number, then choose the nearer one.

  1. Find the two multiples. The number lies between two multiples of 10,000.
    6,540,000 < 6,548,213 < 6,550,000
  2. Check the digit after the rounding digit. Halfway is 5,000. Look at how far the number is past 6,540,000.
    6,548,213 − 6,540,000 = 8,213, and 8,213 is at least 5,000
  3. Round up or down. Five or more means round up. Less than five means round down.
    Round up to 6,550,000

Check. 6,550,000 is a multiple of 10,000 and is the closer one to 6,548,213. ✓

Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.

Worked example 1.3.2
Round 2,497,650 to the nearest 100,000.

Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.

Method. Find the two multiples of 100,000 either side of the number, then choose the nearer one.

  1. Find the two multiples. The number lies between two multiples of 100,000.
    2,400,000 < 2,497,650 < 2,500,000
  2. Check the digit after the rounding digit. Halfway is 50,000. Look at how far the number is past 2,400,000.
    2,497,650 − 2,400,000 = 97,650, and 97,650 is at least 50,000
  3. Round up or down. Five or more means round up. Less than five means round down.
    Round up to 2,500,000

Check. 2,500,000 is a multiple of 100,000 and is the closer one to 2,497,650. ✓

Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.

Worked example 1.3.3
Round 3,499,999 to the nearest 1,000,000.

Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.

Method. Find the two multiples of 1,000,000 either side of the number, then choose the nearer one.

  1. Find the two multiples. The number lies between two multiples of 1,000,000.
    3,000,000 < 3,499,999 < 4,000,000
  2. Check the digit after the rounding digit. Halfway is 500,000. Look at how far the number is past 3,000,000.
    3,499,999 − 3,000,000 = 499,999, and 499,999 is less than 500,000
  3. Round up or down. Five or more means round up. Less than five means round down.
    Round down to 3,000,000

Check. 3,000,000 is a multiple of 1,000,000 and is the closer one to 3,499,999. ✓

Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.

Watch out

Do not stop at the rounding digit. To round 3,499,999 to the nearest 1,000,000, look at the 499,999 part: it is less than 500,000, so the answer is 3,000,000, not 4,000,000. Also, a digit of 5 rounds up, even when it is the last digit: 1,250 to the nearest 100 is 1,300.

Exercise 1C  Rounding

Non-calculator
1
FRound 4,736 to the nearest 10.
4,7304,7404,735halfway4,736
2
FRound 58,265 to the nearest 100.
3
FRound each number to the nearest 1,000.
(a)362,489
(b)84,650
(c)1,907,500
4
FRound each number to the nearest 10,000.
(a)2,753,600
(b)6,473,000
(c)9,996,500
5
FRound each number to the nearest 100,000.
(a)5,249,999
(b)1,640,000
(c)3,720,000
6
FRound each number to the nearest 1,000,000.
(a)7,321,000
(b)8,600,000
(c)4,999,999
7
PSA football stadium holds 45,786 people. Round this to the nearest 1,000 to estimate how many people it holds.
8
RWhich whole numbers round to 4,000,000 to the nearest 1,000,000? Write the smallest and the largest.
3,000,0005,000,0004,000,000smallest 3,500,000largest 4,499,999
9
PSThere are 86,400 seconds in a day. Round this to the nearest 1,000 to estimate.
10
RAli says 5,449,999 rounds to 6,000,000 to the nearest 1,000,000. Is he right? Explain your answer.
11
PSRound −6,548 to the nearest 1,000.
−7,000−6,000
12
RWhat is the smallest whole number that rounds to 3,000,000 to the nearest 100,000 and also rounds to 3,000,000 to the nearest 1,000,000?
1.4

Negative numbers

Year 6
Use negative numbers in context (temperature, height, depth), Count backwards and forwards through zero, Order negative numbers
Video lesson: Negative numbersComing soon

A negative number is a number less than zero. We write a minus sign in front of it, so −4 is “negative four”. Negative numbers are used every day. A thermometer shows −3 °C on a cold winter morning, a submarine can be at −120 m below sea level, and a lift can go down to the basement at −3.

On a number line, numbers get bigger as you move to the right and smaller as you move to the left. So −10 is further left than −4, which means −10 is less than −4. Zero is neither positive nor negative.

−6−5−4−3−2−101234coldestwarmest
A number line that crosses zero. Numbers get smaller to the left, and bigger to the right.
−10−8−6−4−20246810
A thermometer showing −4 °C. The liquid is 4 degrees below zero.

Key ideas

  • A negative number is less than zero. It has a minus sign in front.
  • For negative numbers, the bigger the number after the minus sign, the smaller the number: −9 is less than −3.
  • To count down through zero, go 3, 2, 1, 0, −1, −2, −3…
  • For a temperature change, move up for a rise and down for a fall. Count through zero when you cross it.
  • Zero is neither positive nor negative.
Worked example 1.4.1
Put these numbers in order, from the smallest: −4, 2, −10, 0, 7.

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Put the negative numbers first, then zero, then the positive numbers, all in the right order.

  1. Split the numbers into groups. Negative numbers are less than zero, and positive numbers are greater than zero.
    negative: −10, −4; zero; positive: 2, 7
  2. Order the negative numbers. For negatives, the bigger number after the minus sign is the smaller number: −10 is less than −4.
    −10, −4
  3. Write the whole list from the smallest. Put the smallest number first.
    −10, −4, 0, 2, 7

Check. Each number is in the right place: −10 < −4 < 0 < 2 < 7. ✓

Watch out. A bigger minus number is colder. −10 °C is colder than −4 °C, even though 10 is bigger than 4.

Worked example 1.4.2
At midday the temperature is 3 °C. It falls by 7 degrees. What is the new temperature?

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Start at the first number and move 7 steps down. Count through zero if you need to.

  1. Start at the first number. Find it on the number line or thermometer.
    Start at 3 °C
  2. Move 7 steps down. Count one step at a time. Each step changes the number by 1.
    3 − 7
  3. Write the new number.
    −4 °C

Check. 3 − 7 = −4. Going back: −4 + 7 = 3. ✓

Watch out. Going down past zero makes the number negative. Count through zero, do not stop at zero.

Worked example 1.4.3
Which is colder, −12 °C or −8 °C?

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Both numbers are negative, so the one with the bigger number after the minus sign is colder.

  1. Find where each number is on a thermometer. Start at zero and count down.
    −12 is 12 steps below zero. −8 is 8 steps below zero.
  2. Compare the two numbers. Further below zero means colder.
    −12 is further below zero than −8.
  3. Write the answer.
    −12 °C is colder

Check. Count down from zero: 8 steps reaches −8, and 12 steps reaches −12. ✓

Watch out. Do not choose the number that looks bigger: 12 is bigger than 8, but the temperature is colder.

Watch out

Do not think that −9 is bigger than −3 because 9 is bigger than 3. With negative numbers, the bigger number after the minus sign is the colder, smaller number. Also, zero is not a negative number, so 0 is greater than every negative number.

Exercise 1D  Negative numbers

Non-calculator
1
FWrite the number that is 1 less than zero.
2
FWrite the missing numbers: 3, 2, 1, 0, __, __, __
3
FPut these temperatures in order, from the coldest to the warmest: −5 °C, 3 °C, −9 °C, 0 °C.
4
FWhich is warmer, −2 °C or −6 °C?
5
FWhat temperature does this thermometer show?
−10−8−6−4−20246810
6
FStart at −7 and count up 4. Where do you stop?
−8−7−6−5−4−3−2−10startstop
7
FWork out each answer.
(a)What is 3 more than −5?
(b)What is 4 less than 1?
(c)What is 6 less than −1?
8
FWrite the missing numbers on the number line.
−6−5?−3−2?012
9
FPut these numbers in order, from the largest to the smallest: −15, −3, −20, 0, 4.
10
RBen says, “−9 is greater than −3, because 9 is greater than 3.” Explain why Ben is wrong.
11
PSA submarine is at −120 m. It rises 45 m. What is its new depth? Write your answer with its unit.
12
RAt 6 pm the temperature was −4 °C. By midnight it had fallen by 5 degrees. By 6 am it had risen by 9 degrees. What was the temperature at 6 am?
1.5

Intervals across zero

Year 6
Calculate intervals across zero, Find the difference between a negative and a positive number, Use intervals in temperatures and heights
Video lesson: Intervals across zeroComing soon

An interval is the gap between two numbers. On a number line we can count the steps. When the two numbers are on opposite sides of zero, the easiest way is to count the steps from each number to zero, and then add the two parts.

From −6 to 5: there are 6 steps from −6 up to zero, and 5 steps from zero up to 5. The interval is 6 + 5 = 11.

When both numbers are on the same side of zero, you do not cross zero. Subtract the smaller distance from the bigger distance. From −9 to −4 there are 9 steps to zero and 4 steps to zero, so the interval is 9 − 4 = 5.

−6−5−4−3−2−101234511 steps
The interval from −6 to 5 is 11 steps: 6 steps to zero and 5 more steps to 5.

Key ideas

  • An interval is the number of steps between two numbers.
  • Numbers on opposite sides of zero: add the steps to zero from each number.
  • Numbers on the same side of zero: subtract the smaller distance from the bigger distance.
  • A rise in temperature is the interval from the lower temperature up to the higher one.
  • Check by counting on from the first number to the second, one step at a time.
Worked example 1.5.1
Work out the interval from −8 to 5.

Key words. An interval is the difference between two numbers. To find an interval across zero, count the steps from one number to zero, then from zero to the other number, and add the two parts together.

Method. Find the number of steps between the two numbers. If the numbers are on opposite sides of zero, add the steps to zero.

  1. Count the steps from −8 up to zero. Zero is the middle of the number line.
    8 steps
  2. Count the steps from zero up to 5. Keep counting the same way, one step at a time.
    5 steps
  3. Add the two parts. The total number of steps is the interval.
    8 + 5 = 13

Check. Count on from −8 to 5 one step at a time: you pass zero and count 13 steps in total. ✓

Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.

Worked example 1.5.2
Work out the interval from −9 to −4.

Key words. An interval is the difference between two numbers. To find an interval across zero, count the steps from one number to zero, then from zero to the other number, and add the two parts together.

Method. Find the number of steps between the two numbers. If the numbers are on opposite sides of zero, add the steps to zero.

  1. Find the distance of each number from zero. Both numbers are negative, so use the size of each number without its minus sign.
    −9 is 9 from zero, and −4 is 4 from zero
  2. Subtract the smaller distance from the bigger one. Both numbers are on the same side of zero.
    9 − 4 = 5
  3. Write the interval.
    5

Check. Count up from −9 to −4: that is 5 steps. ✓

Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.

Worked example 1.5.3
The temperature rises from −7 °C to 3 °C. By how many degrees does it rise?

Key words. An interval is the difference between two numbers. To find an interval across zero, count the steps from one number to zero, then from zero to the other number, and add the two parts together.

Method. Find the number of steps between the two numbers. If the numbers are on opposite sides of zero, add the steps to zero.

  1. Count the steps from −7 up to zero. Zero is the middle of the number line.
    7 steps
  2. Count the steps from zero up to 3. Keep counting the same way, one step at a time.
    3 steps
  3. Add the two parts. The total number of steps is the interval.
    7 + 3 = 10

Check. Count on from −7 to 3 one step at a time: you pass zero and count 10 steps in total. ✓

Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.

Worked example 1.5.4
A lift starts at the basement, which is floor −3. It goes up to floor 12. Ground floor is 0. How many floors does it go up?

Key words. An interval is the difference between two numbers. To find an interval across zero, count the steps from one number to zero, then from zero to the other number, and add the two parts together.

Method. Find the number of steps between the two numbers. If the numbers are on opposite sides of zero, add the steps to zero.

  1. Count the steps from −3 up to zero. Zero is the middle of the number line.
    3 steps
  2. Count the steps from zero up to 12. Keep counting the same way, one step at a time.
    12 steps
  3. Add the two parts. The total number of steps is the interval.
    3 + 12 = 15

Check. Count on from −3 to 12 one step at a time: you pass zero and count 15 steps in total. ✓

Watch out. Do not subtract two negative numbers in the same way as positive numbers. Count the steps.

Watch out

Do not subtract when the numbers are on opposite sides of zero. From −4 to 6 is not 6 − 4 = 2. It is 4 + 6 = 10 steps. Also do not forget the step through zero: count it when you cross zero.

Exercise 1E  Intervals across zero

Non-calculator
1
FWork out the interval between each pair of numbers.
(a)from −4 to 6
(b)from −7 to 2
(c)from −12 to −5
(d)from −3 to 8
2
FWork out the interval from −10 to 4.
3
PSAt 6 am the temperature was −6 °C. At noon it was 4 °C. By how many degrees did it rise?
4
PSThe table shows the temperature at noon for five days. Work out the interval between the highest and the lowest temperature.
DayTemperature at noon
Monday−3 °C
Tuesday5 °C
Wednesday−8 °C
Thursday2 °C
Friday−1 °C
5
FHow many steps are there from −5 to 3 on this number line?
−6−5−4−3−2−101234startend
6
FHow many whole numbers are there from −4 to 5, including both of them?
7
PSA submarine is at −120 m. It rises to −35 m. How far does it rise?
8
PSA valley is 85 m below sea level. A hill is 140 m above sea level. Work out the interval from the bottom of the valley to the top of the hill.
9
RWhich interval is longer, from −3 to 8 or from −10 to −2? Explain your answer.
10
PSThe interval from −9 to a positive number is 14. What is the positive number?
11
RAlways, sometimes or never? The interval between a negative number and a positive number is bigger than the positive number. Explain your answer.
12
RThe interval between a negative number and a positive number is 17. The positive number is 3 more than the size of the negative number, without its minus sign. What are the two numbers?
1.6

Solving number and practical problems

Year 6
Solve number and practical problems using place value, rounding and negative numbers, Choose between an exact answer and an estimate
Video lesson: Solving number and practical problemsComing soon

In a problem, first find the numbers you need and the operation to use. Read the question carefully. Does it ask for an exact answer, or an estimate? If it says “to the nearest”, you must round. If it says “how many more”, you need to subtract. If it says “where is it now” with negative numbers, count the steps.

Always write the units with your answer: people, metres, degrees or pounds. Then check that your answer is sensible. Is it about the size you expected?

Key ideas

  • Read the question twice. Note the numbers and the words that tell you the operation.
  • “To the nearest” means round. Choose the place value given in the question.
  • “How many more” or “how much bigger” means subtract.
  • For a change below zero, count the steps through zero.
  • Write the units, and check whether an estimate or an exact answer is needed.
Worked example 1.6.1
A town has 2,376,540 people. Round this to the nearest 100,000 to estimate the population.

Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.

Method. Find the two multiples of 100,000 either side of the number, then choose the nearer one.

  1. Find the two multiples. The number lies between two multiples of 100,000.
    2,300,000 < 2,376,540 < 2,400,000
  2. Check the digit after the rounding digit. Halfway is 50,000. Look at how far the number is past 2,300,000.
    2,376,540 − 2,300,000 = 76,540, and 76,540 is at least 50,000
  3. Round up or down. Five or more means round up. Less than five means round down.
    Round up to 2,400,000

Check. 2,400,000 is a multiple of 100,000 and is the closer one to 2,376,540. ✓

Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.

Worked example 1.6.2
A submarine is at −150 m. It rises 90 m. Where is it now? Give your answer with its unit.

Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.

Method. Start at the first number and move 90 steps up. Count through zero if you need to.

  1. Start at the first number. Find it on the number line or thermometer.
    Start at −150 m
  2. Move 90 steps up. Count one step at a time. Each step changes the number by 1.
    −150 + 90
  3. Write the new number.
    −60 m

Check. −150 + 90 = −60. Going back: −60 − 90 = −150. ✓

Watch out. Going down past zero makes the number negative. Count through zero, do not stop at zero.

Worked example 1.6.3
A museum had 3,140,000 visitors last year. This year it had 480,000 more visitors. How many visitors did it have this year?

Method. This year is last year plus the extra visitors. Add them together.

  1. Find the extra visitors. This year it had 480,000 more than last year.
    480,000 more
  2. Add them to last year. Add the extra visitors to last year's number.
    3,140,000 + 480,000 = 3,620,000
  3. Check. Subtract the extra visitors to get back to last year.
    3,620,000 − 480,000 = 3,140,000

Check. 3,620,000 − 480,000 = 3,140,000. ✓

Watch out. Read carefully: “more” means add, and “less” means subtract.

Watch out

When a question asks for the answer to the nearest 100,000, do not give the exact answer. Also, do not round the numbers before you add them if the question asks for an exact answer. Round only at the end when you are asked to.

Exercise 1F  Problems with place value, rounding and negative numbers

Non-calculator
1
FA stadium has 48,650 seats. Round this to the nearest 1,000 to estimate how many seats there are.
2
PSIn January a factory made 3,450,000 bolts. In February it made 3,725,000 bolts. How many more bolts did it make in February?
3
PSA deep-sea robot is at −2,450 m. It rises 1,800 m. Where is it now? Give your answer with its unit.
4
FA library has 1,286,400 books. Round this to the nearest 10,000.
5
PSA charity raised £1,240,000. Round this to the nearest £100,000.
6
FHow many 100,000s make 2,500,000?
7
PSThree towns have these populations: Ashford 2,400,000; Brindley 1,850,000; Carlton 3,250,000.
0500,0001,000,0001,500,0002,000,0002,500,0003,000,0003,500,000Ashford2,400,000Brindley1,850,000Carlton3,250,000
(a)Which town has the largest population?
(b)How many more people live in the largest town than in the smallest town?
8
RMia writes that 2,500,000 is the same as 2,500 thousand. Is she right? Explain your answer.
9
FWrite the number that is 1,000 less than 1,000,000.
10
PSThere are 3,600 seconds in one hour. Round this to the nearest 1,000.
11
FWhat is 100,000 less than 3,000,000?
12
RA 7-digit number rounds to 8,000,000 to the nearest 1,000,000, and rounds to 7,990,000 to the nearest 10,000. What is the smallest such number?

Chapter summary

SkillMethod
Place valueEach digit is worth its digit times its column value. Columns go ten times bigger from right to left: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions.
Reading and writingSplit the number into groups of three from the right. Read each group and say its name: million, thousand, or the ones group.
Comparing and orderingCompare the digits from the left. The first column where the digits differ decides. Use < and > with the open end facing the bigger number.
RoundingFind the two multiples either side of the number. Look at the digit after the rounding digit: 5 or more rounds up, less than 5 rounds down. Halfway rounds up.
Negative numbersA negative number is less than zero. The bigger the number after the minus sign, the smaller the number. Count through zero when you cross it.
Intervals across zeroOpposite sides of zero: add the steps to zero from each number. Same side of zero: subtract the smaller distance from the bigger distance.
ProblemsFind the numbers and the operation. “To the nearest” means round. “How many more” means subtract. Write the unit and check the answer makes sense.

Can you…?

  • I can read and write numbers up to 10,000,000 in figures and words.
  • I can find the value of any digit in a number up to 10,000,000.
  • I can compare and order numbers up to 10,000,000 using the signs < and >.
  • I can make the largest and smallest numbers from a set of digit cards.
  • I can round any whole number to the nearest 10, 100, 1,000, 10,000, 100,000 or 1,000,000.
  • I can order negative numbers and count through zero on a number line or thermometer.
  • I can work out the interval between two numbers, including numbers on opposite sides of zero.
  • I can solve number and practical problems using place value, rounding and negative numbers.

Key terms

Digit
One of the symbols 0 to 9 used to write a number.
Column
The place a digit sits in, such as the hundreds or the ten thousands column.
Value
What a digit is worth. The value is the digit times the value of its column.
Period
A group of three columns, such as the thousands or the millions.
Ten million
The number 10,000,000. It has eight digits and seven zeros.
Round
Make a number simpler by writing it to the nearest 10, 100, 1,000 and so on.
Degree of accuracy
The place value a number is rounded to, such as the nearest 1,000.
Halfway
The number exactly in the middle of two multiples. Halfway rounds up.
Negative number
A number less than zero, written with a minus sign, such as −5.
Positive number
A number greater than zero.
Interval
The number of steps, or the difference, between two numbers on a number line.
Thermometer
A scale that shows temperature. Numbers below zero are shown with a minus sign.
Compare
Decide which of two numbers is bigger or smaller, using the signs < and >.
Digit card
A card with one digit on it, used to make numbers.

Exercise 1Z  Check-up

Non-calculator
1
What is the value of the 9 in 5,920,814?
2
Write 3,070,405 in words.
3
Write in figures: four million, sixty thousand and twelve.
4
Write the correct sign, < or >: 7,410,000  ?  7,401,999
5
Write these numbers in order, from the smallest to the largest: 5,020,000;  502,000;  5,200,000;  5,002,000;  5,020,100.
6
Round 8,465,300 to the nearest 100,000.
8,400,0008,500,0008,450,000halfway8,465,300
7
Round 6,043,250 to the nearest 1,000.
8
The temperature is −4 °C. It falls by 6 degrees. What is the new temperature?
9
Work out the interval from −12 to 9.
10
A diver is at −235 m. She rises 80 m. Where is she now? Write your answer with its unit.

Total 20 marks. Allow about 20 minutes.

Exercise 1X  SATs-style questions

Non-calculator
1
FWrite the value of the 4 in 1,438,920.[1 mark]
2
FCircle the largest number:   4,250,000    4,520,000    4,205,000    4,502,000[1 mark]
3
FWrite 6,002,050 in words.[1 mark]
4
FUse the digits 0, 0, 3, 5, 6, 8 and 9, each once, to make the smallest 7-digit number.[2 marks]
5
FRound 3,749,500 to the nearest 1,000.[2 marks]
6
FWrite these numbers in order, from the smallest to the largest: 2,300,000;  230,000;  2,030,000;  2,003,000.[2 marks]
7
FWhich number rounds to 6,000,000 to the nearest 1,000,000?   Tick one box.   A  5,250,000   B  5,480,000   C  6,520,000   D  6,490,000[2 marks]
8
FAt 6 am the temperature was −7 °C. By noon it was 5 °C. How many degrees did the temperature rise?   Show your working.[2 marks]
9
RSam says, “−14 is greater than −9, because 14 is bigger than 9.” Explain why Sam is wrong.[3 marks]
10
PSA diver starts at −240 m. She rises 135 m. Then she dives 60 m. Where is she now?   Show your working.[4 marks]
Chapter 2: The four operations and order of operations →

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