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SATs practice · Year 6Chapter 1 · Place value and the four operations
Year 6 · Chapter 1

Place value and the four operations

By the end of this chapter you will be able to

  • Find the value of a digit in a number up to 10,000,000
  • Round a number to the nearest 10, 100, 1000 or 100,000
  • Add and subtract large numbers using a column method
  • Count through zero with negative numbers in a context
  • Work out a calculation in the correct order of operations

You should already know

  • Place value to 1,000,000 and rounding (Year 5 maths)
  • Adding and subtracting with the column method (Year 5 maths)

Where this leads

  • Multiplying and dividing whole numbers (Chapter 2 of this book)

KS2 SATs practice for arithmetic (Year 6). Original questions in the style of the national tests. Content: Read, write, order and compare whole numbers up to 10,000,000 and determine the value of each digit (Year 5-6), Count forwards and backwards in powers of 10 (Year 5), Interpret negative numbers in context, and count forwards and backwards through zero (Year 6), Round any whole number to a required degree of accuracy (Year 5-6), Add and subtract whole numbers with more than four digits using the formal columnar method (Year 5), Add and subtract mentally with large numbers (Year 5-6), Use the order of operations: brackets, then division and multiplication, then addition and subtraction (Year 6), Use estimation and rounding to check answers; solve problems combining these (Year 5-6).

Exercise P  Do you remember?

Non-calculator
1
FWrite 4 thousands, 2 hundreds and 9 ones as a number.
2
FWork out 2,718 + 1,000.
3
FWork out 5,000 − 1,234.
4
FRound 46 to the nearest 10.
5
FWhat is 7 × 8?
1.1

Place value to 10,000,000

Year 6
Read, write, order and compare whole numbers up to 10,000,000 and determine the value of each digit, Count forwards and backwards in powers of 10
Video lesson: Place value to 10,000,000Coming soon
Practice material. Non-calculator practice. This book is practice in the style of the Year 6 arithmetic test. It is not the official test, and no official test item is used. Do every question without a calculator. Marks and timings are a practice guide only.
Story. The school library counts its books by hundreds of thousands. The head teacher writes 4,702,815 in her report and asks the class to say what the 7 is worth.

Rule

Every column is worth ten times the column on its right: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions and ten millions.

The value of a digit is the digit multiplied by its column. In 4,702,815 the 7 is in the hundred thousands column, so it is worth 7 × 100,000 = 700,000.

Multiplying by 10 moves every digit one column to the left. For the full work on place value, read Year 6 maths, chapter 1.

Key ideas

  • Each column is ten times the column on its right.
  • The value of a digit is the digit times its column value.
  • Compare two numbers from the left: the first column that differs decides.
  • Multiplying by 10, 100 or 1000 shifts the digits to the left.
Worked example 1.1.1
What is the value of the 7 in 4,702,815?

Method. Find the column of the 7, then multiply it by the column value.

  1. Find the column of the digit. Count from the right: this is the hundred thousands column.
    digit 7 in the hundred thousands column
  2. Multiply the digit by the column value. Each column is worth ten times the column on its right.
    7 × 100,000 = 700,000
  3. Write the answer.
    700,000

Check. The 7 is in the hundred thousands column, and 700,000 + 4,002,815 = 4,702,815. ✓

Watch out. The 7 is not worth 7. Its column makes it worth seven hundred thousand.

Worked example 1.1.2
Write the number: 3 millions, 40 thousands and 6 hundreds.

Method. Write each part in its column, then add them.

  1. Write the millions. 3 millions is 3,000,000.
    3,000,000
  2. Write the thousands. 40 thousands is 40,000.
    3,000,000 + 40,000
  3. Write the hundreds. 6 hundreds is 600.
    3,000,000 + 40,000 + 600 = 3,040,600
  4. Write the answer.
    3,040,600

Check. Count the zeros: the hundreds, tens and ones columns need zeros where nothing is written. ✓

Watch out. Leave the tens and ones as zeros if the question does not mention them.

Worked example 1.1.3
Put these numbers in order, smallest first: 5,091,000; 509,100; 5,190,000; 509,010.

Method. Compare the numbers from the left. Count the digits first.

  1. Count the digits. Two numbers have seven digits and two have six.
    509,100 and 509,010 have six digits
  2. Compare the six-digit numbers. Both start 509, then 1 is bigger than 0.
    509,010 < 509,100
  3. Compare the seven-digit numbers. Both start with 5 million. In the hundred thousands column 0 is less than 1, so 5,091,000 is smaller.
    5,091,000 < 5,190,000
  4. Write the answer.
    509,010, 509,100, 5,091,000, 5,190,000

Check. Each number is smaller than the one after it. ✓

Watch out. 509,010 has six digits, so it is smaller than 5,091,000, which has seven.

Exercise 1A  Practice: place value to ten million

Non-calculator
1
FWhat is the value of the 9 in 3,904,216?
2
FWhat is the value of the 3 in 83,457?
3
FWrite the number: 3 millions, 40 thousands and 6 hundreds.
4
FCount on in steps of 100,000 from 3,600,000. What number is 3 steps on?
5
FWhat is 10 times 46,250?
6
FWhat is 1,000 times 3,072?
7
FWhich number is 1,000,000 more than 7,250,400?
8
PSWrite the number with 9 in the ten thousands, 7 in the thousands, 6 in the tens and 3 in the ones, and 0 in every other column.
9
FFill the box with <, > or =: 6,482,000 ☐ 6,428,000
10
PSWhich number has a 4 in the hundred thousands column? Choose one: 1,452,000, 3,640,000, 4,013,000 or 2,345,678.
11
PSWrite the largest number you can make with the digits 9, 0, 4, 4, 1, 7 and 2, using each digit once.
12
RSam says: "In 5,555,555 every digit is worth the same." Is he right? Explain.
1.2

Negative numbers in context

Year 6
Interpret negative numbers in context, and count forwards and backwards through zero (Year 6)
Video lesson: Negative numbers in contextComing soon
Story. On a cold morning the thermometer outside the bakery reads -4°C. By lunchtime it has risen by 3 degrees.

Rule

A negative number is less than zero. On a number line, -1 is one step below 0 and -5 is five steps below.

Counting up moves towards zero and then into the positives. Counting down moves further below zero.

The bigger the negative number ignoring its minus sign, the smaller the number: -9 is less than -2.

Key ideas

  • Negative numbers are below zero: -1, -2, -3 ...
  • Going up from -4 by 3 gives -1: the count stops at -1, not at 1.
  • Less than means further left on the number line.
  • Counting up from a negative number moves towards zero.
Worked example 1.2.1
The temperature is -4°C and rises by 3°C. What is the new temperature?

Method. Count up 3 steps from -4 on a number line.

  1. Start at -4. The thermometer starts at -4°C.
    -4°C
  2. Count up 3 steps. Each step adds 1: -3, -2, -1.
    -4 → -3 → -2 → -1
  3. Read the answer. Three steps up from -4 is -1.
    -4 + 3 = -1
  4. Write the answer.
    -1°C

Check. -1 + 1 = 0, so -1 is one step below zero. ✓

Watch out. Do not go up past zero: -4 + 3 is -1, not 1.

Worked example 1.2.2
A diver is 12 m below the surface, which we write as -12 m. She swims up 9 m. Where is she now?

Method. Move up 9 metres from -12 by counting up through the negatives.

  1. Start at -12. The diver is 12 m below the surface.
    -12 m
  2. Count up 9 metres. Going up adds to the number: -12 + 9 = -3.
    -12 + 9 = -3
  3. Read the answer in the context. She is 3 m below the surface.
    -3 m
  4. Write the answer.
    -3 m

Check. She is still below the surface, so the answer is negative. ✓

Worked example 1.2.3
Which is colder: -9°C or -2°C?

Method. The further a negative number is from zero, the colder it is.

  1. Compare the two temperatures. Both are below zero.
    -9 and -2
  2. Think about the number line. -9 is further left than -2, so it is less.
    -9 < -2
  3. Say which is colder. The smaller temperature is colder.
    -9°C is colder
  4. Write the answer.
    -9°C

Check. -9 is less than -2. ✓

Watch out. -9 is less than -2, even though 9 is bigger than 2.

Exercise 1B  Practice: negative numbers in context

Non-calculator
1
FWhat is 2 more than -5?
2
FCount from -3 up to 2. How many steps is that?
3
FWhich temperature is the coldest? Choose one: -7°C, -2°C, 0°C or 3°C.
4
FThe temperature at night is -6°C. What is it after it gets 6 degrees warmer?
5
FWrite the number that is 4 less than 1.
6
FThe temperature is 1°C at noon and falls by 4 degrees. What is the temperature now?
7
PSA bank balance is -£35. A payment of £60 goes in. What is the new balance?
8
PSA diver is at -12 m. He swims up 9 m. Where is he?
9
PSThe temperature falls from 3°C to -5°C. How many degrees does it fall?
10
PSA lift starts in the basement at floor -2. It goes up 5 floors. Which floor is it on now?
11
RWhich is greater: -15 or -9? Choose one: -15 or -9.
12
RAli says -2 is bigger than -7 because 2 is smaller than 7. Is he right? Explain.
1.3

Rounding and estimation

Year 6
Round any whole number to a required degree of accuracy (Year 5-6), Use estimation and rounding to check answers (Year 5-6)
Video lesson: Rounding and estimationComing soon
Story. A school choir sells 2,465 tickets at a concert. The newspaper says "about 2,500 people came", because the number was rounded to the nearest 100.

Rule

To round to a given place, look at the digit just to the right of that column. If it is 5 or more, round up. If it is 4 or less, round down.

Estimating means rounding each number first, then working with the rounded numbers. It shows whether your answer is about right.

Key ideas

  • Round up when the digit to the right is 5, 6, 7, 8 or 9.
  • Round down when that digit is 0, 1, 2, 3 or 4.
  • Estimate by rounding every number to the same place value.
  • An estimate tells you the answer is about right, not the exact answer.
Worked example 1.3.1
Round 36,479 to the nearest 1,000.

Method. Find the two thousands either side, then look at the hundreds digit.

  1. Find the two multiples of the rounding amount either side of the number. The number lies between 36,000 and 37,000.
    36,000 < 36,479 < 37,000
  2. Look at the digit just to the right of the rounding column. It shows which multiple is nearer.
    digit 4
  3. Round to the nearest thousand. The digit is 4, which is less than 5, so round down.
    36,479 rounds to 36,000
  4. Write the answer.
    36,000

Check. 36,000 is 479 less than 36,479, and 37,000 is 521 more. ✓

Watch out. Look at the hundreds digit (4), not the thousands digit.

Worked example 1.3.2
Round 2,850,000 to the nearest 100,000.

Method. Look at the digit to the right of the hundred thousands column.

  1. Find the two multiples of the rounding amount either side of the number. The number lies between 2,800,000 and 2,900,000.
    2,800,000 < 2,850,000 < 2,900,000
  2. Look at the digit just to the right of the rounding column. It shows which multiple is nearer.
    digit 5
  3. Round to the nearest hundred thousand. The digit is 5, which is 5 or more, so round up.
    2,850,000 rounds to 2,900,000
  4. Write the answer.
    2,900,000

Check. 2,850,000 is halfway between 2,800,000 and 2,900,000, and halves round up. ✓

Worked example 1.3.3
Estimate 4,912 + 3,087 by rounding each number to the nearest thousand.

Method. Round both numbers first, then add the rounded numbers.

  1. Round the first number. Nearest thousand: 4,912 is nearer to 5,000.
    4,912 → 5,000
  2. Round the second number. Nearest thousand: 3,087 is nearer to 3,000.
    3,087 → 3,000
  3. Add the rounded numbers. The estimate is about 8,000. The exact answer is 8,000 too.
    5,000 + 3,000 = 8,000
  4. Write the answer.
    8,000

Check. The exact answer is 7,999, which is close to the estimate of 8,000. ✓

Watch out. An estimate can be close but not exact: 7,999 and 8,000 differ by 1.

Exercise 1C  Practice: rounding and estimating

Non-calculator
1
FRound 8,463 to the nearest 100.
2
FRound 58,241 to the nearest 1,000.
3
FRound 672 to the nearest 10.
4
FRound 3,465,000 to the nearest 100,000.
5
FRound 9,999,997 to the nearest million.
6
FRound 412,000 to the nearest 10,000.
7
FRound 15,608 to the nearest 1,000.
8
PSEstimate 48,207 - 19,586 by rounding each number to the nearest 10,000.
9
PSA school has 2,465 pupils. Round the number to the nearest 100, and write the answer.
10
PSSam works out 3,812 + 5,190 = 8,002. Round each number to the nearest 1,000 to check. Is the answer sensible? Write Yes or No.
11
RRound 1,450 to the nearest 100, then round that answer to the nearest 1,000. Then round 1,450 to the nearest 1,000 in one step. Are the two final answers the same? Write both final answers.
12
RExplain why 8,450 rounds to 8,000 to the nearest 1,000 but to 8,500 to the nearest 100.
1.4

Adding and subtracting large numbers

Year 6
Add and subtract whole numbers with more than four digits using the formal columnar method (Year 5), Add and subtract mentally with large numbers (Year 5-6)
Video lesson: Adding and subtracting large numbersComing soon
Story. In one year a city museum has 2,345,678 visitors. The next year it has 1,987,654. The manager adds the two totals in a column, carrying where needed.

Rule

Line the numbers up so that each digit is in its own column. Start at the ones column on the right.

In addition, if a column makes ten or more, write the ones digit and carry the tens digit to the next column.

In subtraction, if the top digit is smaller, borrow 10 from the column on its left.

Mental methods can split a number: 2,400,000 + 300,000 is 2,700,000 because 24 + 3 = 27 hundred thousand.

Key ideas

  • Line up the place values before you start.
  • Carry in addition when a column makes 10 or more.
  • Borrow in subtraction when the top digit is too small.
  • Check a subtraction by adding the answer back to the smaller number.
Worked example 1.4.1
Work out 4,718,260 + 2,396,845.

Method. Add column by column from the right, carrying where needed.

  1. Add the ones column. Write the ones digit of the total in this column, and carry the tens digit to the next column.
    0 + 5 + 0 = 5, write 5
  2. Add the tens column. Write the ones digit of the total in this column, and carry the tens digit to the next column.
    6 + 4 + 0 = 10, write 0, carry 1
  3. Add the hundreds column. Write the ones digit of the total in this column, and carry the tens digit to the next column.
    2 + 8 + 1 = 11, write 1, carry 1
  4. Add the thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.
    8 + 6 + 1 = 15, write 5, carry 1
  5. Add the ten thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.
    1 + 9 + 1 = 11, write 1, carry 1
  6. Add the hundred thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.
    7 + 3 + 1 = 11, write 1, carry 1
  7. Add the millions column. Write the ones digit of the total in this column, and carry the tens digit to the next column.
    4 + 2 + 1 = 7, write 7
  8. Write the answer.
    7,115,105

Check. Check the other way round: 2,396,845 + 4,718,260 = 7,115,105. ✓

Watch out. Do not forget to carry from one column to the next.

Worked example 1.4.2
Work out 6,004,517 - 2,398,649.

Method. Subtract column by column from the right, borrowing where needed.

  1. Subtract the ones column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.
    7 - 0 = 7, which is less than 9, so borrow 10: 17 - 9 = 8
  2. Subtract the tens column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.
    1 - 1 = 0, which is less than 4, so borrow 10: 10 - 4 = 6
  3. Subtract the hundreds column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.
    5 - 1 = 4, which is less than 6, so borrow 10: 14 - 6 = 8
  4. Subtract the thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.
    4 - 1 = 3, which is less than 8, so borrow 10: 13 - 8 = 5
  5. Subtract the ten thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.
    0 - 1 = -1, which is less than 9, so borrow 10: 9 - 9 = 0
  6. Subtract the hundred thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.
    0 - 1 = -1, which is less than 3, so borrow 10: 9 - 3 = 6
  7. Subtract the millions column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.
    5 - 2 = 3
  8. Write the answer.
    3,605,868

Check. Check: 3,605,868 + 2,398,649 = 6,004,517. ✓

Watch out. Borrowing across several zeros is the hardest case: check each column.

Worked example 1.4.3
Work out mentally: 2,400,000 + 300,000.

Method. Add the hundred thousands, not the individual digits.

  1. Split the numbers. 2,400,000 is 24 hundred thousand and 300,000 is 3 hundred thousand.
    24 + 3 = 27 hundred thousand
  2. Put it back together. Twenty-seven hundred thousand is 2,700,000.
    2,700,000
  3. Write the answer.
    2,700,000

Check. 2,700,000 - 300,000 = 2,400,000. ✓

Exercise 1D  Practice: adding and subtracting large numbers

Non-calculator
1
FWork out 1,256,340 + 2,118,095.
2
FWork out 7,000,000 - 3,250,000.
3
FWork out 45,678 + 9,876.
4
FWork out 100,000 - 37,426.
5
FWork out mentally: 3,004,000 + 996,000.
6
FWork out mentally: 8,500,000 - 1,000,000.
7
FWork out mentally: 25,000 + 4,900.
8
PSA museum had 2,345,678 visitors in one year and 1,987,654 the next. How many visitors were there in total?
9
PSA town has 3,450,000 people. 1,268,500 of them live in the north. How many people live elsewhere in the town?
10
PSSam has £10,000 in a savings account. He spends £3,678 on a bike. How much is left?
11
RExplain how you would check 5,461,000 - 1,234,000 = 4,227,000 without working it out again.
12
RSam says 8,000,000 - 7,999,999 = 2. Is he right? Write the correct answer.
1.5

Order of operations and mixed problems

Year 6
Use the order of operations: brackets, then division and multiplication, then addition and subtraction (Year 6), Solve problems combining these (Year 5-6)
Video lesson: Order of operations and mixed problemsComing soon
Story. Ali buys three packs of six pencils and four single pencils. He writes 3 × 6 + 4 on his paper, and the order he does the sums in matters.

Rule

Work out brackets first. Then do multiplication and division, working from left to right. Then do addition and subtraction, working from left to right.

So 3 + 4 × 5 is 3 + 20 = 23, not 35.

Key ideas

  • Brackets first, then multiply and divide, then add and subtract.
  • Multiplication and division are done before addition and subtraction.
  • Work from left to right when the operations are the same.
  • Check by estimating: the answer should be about right.
Worked example 1.5.1
Work out 3 + 4 × 5.

Method. Multiplication is done before addition.

  1. Multiply first. Do 4 × 5 before adding 3.
    4 × 5 = 20
  2. Add the 3. Now add 3 to 20.
    3 + 20 = 23
  3. Write the answer.
    23

Check. Not (3 + 4) × 5 = 35: the brackets are missing from the question. ✓

Watch out. Without brackets, the multiplication is done first, so the answer is 23.

Worked example 1.5.2
Work out (8 + 4) ÷ 3 × 2.

Method. Brackets first, then division and multiplication from left to right.

  1. Do the brackets. 8 + 4 = 12 inside the brackets.
    (8 + 4) = 12
  2. Divide. Then 12 ÷ 3 = 4.
    12 ÷ 3 = 4
  3. Multiply. Finally 4 × 2 = 8.
    4 × 2 = 8
  4. Write the answer.
    8

Check. Working left to right after the brackets gives the same answer. ✓

Worked example 1.5.3
Work out 20 - 3 × 4 + 6.

Method. Multiply before you add and subtract, and then go from left to right.

  1. Multiply. 3 × 4 = 12 is done first.
    20 - 12 + 6
  2. Subtract. Then 20 - 12 = 8.
    8 + 6
  3. Add. Finally 8 + 6 = 14.
    14
  4. Write the answer.
    14

Check. The multiplication comes before the subtraction and addition. ✓

Watch out. Doing the subtraction first gives 17 × 4 + 6 = 74, which is wrong.

Exercise 1E  Practice: order of operations and problems

Non-calculator
1
FWork out 6 + 3 × 4.
2
FWork out 20 ÷ 4 + 6.
3
FWork out (9 - 3) × 4.
4
FWork out 18 - 12 ÷ 3.
5
FWork out 2 × (5 + 7) - 4.
6
FWork out (12 + 8) ÷ 5.
7
PSAli buys 3 packs of 6 pencils and 4 single pencils. How many pencils does he have?
8
PSA hall has 12 rows of 15 chairs. 18 chairs are taken away. How many chairs are left?
9
PSTickets cost £4 each for 5 adults and £2 each for 6 children. How much do they cost in total?
10
PSPut one pair of brackets in 7 + 5 × 3 so that the answer is 36. Write the calculation.
11
PSPut one pair of brackets in 4 + 3 × 2 + 1 so that the answer is 15. Write the calculation.
12
RSam says 10 - 4 - 3 = 9 because he does 4 - 3 first. Is he right? Explain.
13
RWhich calculation is correct? Choose one: 2 + 3 × 4 = 20, 2 + 3 × 4 = 14 or 2 + 3 × 4 = 24.

Chapter summary

SkillMethod
Value of a digitMultiply the digit by its column: 7 in 4,702,815 is 7 × 100,000 = 700,000.
Comparing numbersLine up the columns and compare from the left; the first column that differs decides.
RoundingLook at the digit to the right of the rounding column: 5 or more rounds up, 4 or less rounds down.
Column additionAdd column by column from the right, carrying tens to the next column.
Column subtractionSubtract column by column from the right, borrowing 10 when the top digit is too small.
Negative numbersCount through zero on a number line; -4 + 3 = -1.
Order of operationsBrackets, then multiply and divide, then add and subtract, working left to right.

Can you…?

  • I can name the value of any digit in a number up to 10,000,000
  • I can round a number to the nearest 10, 100, 1,000, 10,000, 100,000 or million
  • I can add and subtract large numbers using a column method
  • I can count through zero with negative numbers in a real context
  • I can use the order of operations, including brackets, to get the right answer
  • I can estimate by rounding first, and say whether an answer is sensible

Key terms

place value
the value of a digit, which depends on its column
digit
one of the symbols 0 to 9 used to write a number
round
to change a number to the nearest multiple of a given amount
estimate
a sensible, rounded answer that shows whether the exact answer is about right
carry
move ten from one column to the next in addition
borrow
take ten from the next column when subtracting
negative number
a number less than zero, such as -4
order of operations
the rule that says which part of a calculation is done first

Exercise 1Z  Check-up

Non-calculator
1
What is the value of the 8 in 6,280,491?
2
Round 743,526 to the nearest 10,000.
3
Work out 4,805,602 + 1,937,458.
4
Work out 3,000,000 - 1,486,750.
5
The temperature is -8°C. It rises by 11 degrees. What is the new temperature?
6
Work out 9 + 2 × 6 - 4.
7
Work out (14 - 6) × 3 + 2.
8
A cinema has 30 rows of 22 seats. 68 seats are broken. How many seats can be used?
9
Estimate 6,197 + 2,806 by rounding each number to the nearest thousand. Write the estimate.
10
Explain why 7,000 - 5 is not the same as 7,000 - 50 - 5.

Total 18 marks. Allow about 18 minutes.

Exercise 1X  Practice test questions

Non-calculator
1
FWhat is the value of the 8 in 1,580,342?[1 mark]
2
FWrite the number 2 million, 7 thousands and 45.[1 mark]
3
FWhich is the largest number? Choose one: 3,999,999, 4,000,000, 3,900,000 or 3,990,000.[1 mark]
4
FRound 582,500 to the nearest 1,000.[2 marks]
5
FWork out 5,406,219 - 2,870,384.[2 marks]
6
PSWork out 15 + 4 × 6 - 9.[2 marks]
7
PSThe temperature is -3°C at 6 am. It falls by 5 degrees. What is the temperature now?[2 marks]
8
PSA shop sells 180 eggs in one week and 1,250 eggs in the next week. How many eggs does it sell in the two weeks?[2 marks]
9
PSWork out (25 - 9) ÷ 4 + 3.[2 marks]
10
RExplain why the answer to 6,000,000 - 999,999 is more than 5,000,000.[3 marks]
Chapter 2: Multiplication and division →

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