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SATs practice · Year 6Chapter 1 · Number reasoning
Year 6 · Chapter 1

Number reasoning

By the end of this chapter you will be able to

  • Say the value of a digit from its column in a large number
  • Round a number to the nearest 10, 100, 1000 or 10,000 and explain the choice
  • Count through zero and find the difference between negative numbers
  • Use brackets and the order of operations to get the right answer
  • Solve a multi-step word problem and check it with an estimate

You should already know

  • Place value to 1,000,000 and rounding (Year 4 and Year 5 number)
  • The four operations with whole numbers (Year 5 number)

Where this leads

  • Fractions and decimals reasoning (Chapter 2 of this book)

KS2 SATs practice for reasoning (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 find the value of each digit (Year 5-6), Use negative numbers in context and count through zero (Year 6), Round any whole number to a required degree of accuracy (Year 5-6), Use the order of operations, including brackets (Year 6), Solve multi-step problems and decide which operation to use (Year 5-6), Use factors, multiples and primes in problems (Year 5), Use estimation to check whether an answer is sensible, and explain a method (Year 5-6).

Exercise P  Do you remember?

Non-calculator
1
FWrite the value of the 5 in 3,541.
2
FWhat is 1,000 more than 6,850?
3
FWork out 48 x 5.
4
FWrite the next number in this list: 2, 4, 6, 8, __.
5
FIs 27 a multiple of 9? Write yes or no.
1.1

Place value, comparing and rounding

Year 6
Read, write, order and compare whole numbers up to 10,000,000 and find the value of each digit (Year 5-6), Round any whole number to a required degree of accuracy (Year 5-6)
Video lesson: Place value, comparing and roundingComing soon
Practice material. Do this without a calculator. This is practice in the style of the Year 6 maths reasoning test. It is not the official test, and no official test item is used. Marks and timings are a practice guide only.
Story. A festival team counts visitors at the gate. On Saturday the counter shows 4,862,139 visitors, and the report to the council must give the number to the nearest 10,000.

Rule

Each digit has a value that depends on its column. In 4,862,139 the 6 is in the ten thousands column, so it is worth 6 x 10,000 = 60,000.

To round, look at the digit next to the column you are rounding to. 5 or more: round up. 4 or less: round down.

For the full method of place value and rounding, read Year 5 and Year 6 maths, chapter 1.

Key ideas

  • The value of a digit = the digit x the value of its column.
  • Compare numbers by their number of digits first, then by the leading digit.
  • To round to the nearest 10,000, look at the thousands digit.
Worked example 1.1.1
What is the value of the digit 6 in 4,862,139?

Method. Find the column, then multiply the digit by its column value.

  1. Find the column of the digit. The digit 6 is in the ten thousands column.
    ten thousands
  2. Work out the value. The value is the digit times the column value 10,000.
    6 x 10,000 = 60,000
  3. Write the answer.
    60,000

Check. 60,000 = 6 x 10,000. ✓

Watch out. The value of a digit depends on its column, not just on the digit.

Worked example 1.1.2
Round 56,750 to the nearest 1,000.

Method. Look at the digit next to the rounding column.

  1. Find the rounding column and the digit next to it. Rounding to the nearest 1,000 looks at the hundreds digit.
    digit to check = 7
  2. Decide up or down. A 5 or more rounds up; 4 or less rounds down.
    7 is 5 or more, so round up
  3. Write the rounded number. Keep the digits to the left and make the rest zeros.
    57,000
  4. Write the answer.
    57,000

Check. 57,000 is a multiple of 1,000 and is the closest one to 56,750. ✓

Watch out. Do not round a 5 down. A 5 always rounds up.

Worked example 1.1.3
Put these numbers in order, from smallest to largest: 45,000, 4,500, 405,000, 54,000.

Method. Compare the digits, starting from the left.

  1. Count the digits in each number. A number with more digits is larger.
    45,000 has 5 digits, 4,500 has 4 digits, 405,000 has 6 digits, 54,000 has 5 digits
  2. Compare numbers with the same number of digits. Look at the leading digit, then the next one.
    compare from the left
  3. Write them in order. Put them smallest to largest.
    4,500, 45,000, 54,000, 405,000
  4. Write the answer.
    4,500, 45,000, 54,000, 405,000

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

Watch out. 4,500 has four digits and 45,000 has five, so 45,000 is larger.

Watch out

Watch out. The 7 in 700,000 is worth 700,000, not 7. A digit is worth more the further left it is.

Exam tip

Practice guide. Check each rounding answer: it must be a multiple of the rounding unit and close to the original number.

Exercise 1A  Practice: place value, comparing and rounding

Non-calculator
1
FWhat is the value of the digit 3 in 907,314?
2
FWhat is the value of the digit 6 in 2,650,481?
3
FWrite the number: three million, forty thousand and seven. Use commas.
4
FRound 4,862,139 to the nearest 10,000.
5
FRound 56,750 to the nearest 1,000.
6
PSRound 9,996 to the nearest 100.
7
FPut these numbers in order, from smallest to largest: 4,500, 45,000, 405,000, 54,000.
8
FPut these numbers in order, from largest to smallest: 7,063, 1,700, 70,630, 1,070.
9
FWrite the correct sign, < or >: 3,402,871 __ 3,420,871
10
PSAnswer both parts.
(a)(a) What is 1,000 more than 6,495,300?
(b)(b) What is 10,000 less than 6,495,300?
11
RTick the numbers that are greater than 2 million. Options: 1,990,000 | 2,000,000 | 2,004,500 | 1,999,999 | 2,400,000. Write the options you would tick, in the order given.
12
RSam says the 7 in 2,700,000 is worth 7,000. Choose one: Yes, Sam is right, No, the 7 is worth 700,000 or No, the 7 is worth 70.
13
PSUse the digits 4, 1, 6, 0, 8, 2, 9 once each to make the largest seven-digit number you can. Write it.
14
RWrite a number between 4,000 and 5,000 that rounds to 4,500 to the nearest 100.
15
PSRound 7,458 to the nearest 100.
16
RA number rounds to 60,000 to the nearest 10,000. Write the smallest whole number it could be.
1.2

Negative numbers and counting through zero

Year 6
Use negative numbers in context and count through zero (Year 6)
Video lesson: Negative numbers and counting through zeroComing soon
Story. On a winter morning the thermometer outside the shed reads -3 degrees at 7 am. By lunchtime it has risen by 7 degrees, and by midnight it has fallen by 9 degrees.

Rule

Negative numbers are less than zero. On a number line, numbers get smaller as you move left.

To find the change from a negative number, count the steps across zero. From -3 up 7 is -3 + 7 = 4.

Key ideas

  • -3 is smaller than -1, because it is further left on the number line.
  • Going up by 7 from -3 passes through zero and ends at 4.
  • The difference between 9 and -4 is 13, because you count 4 steps to zero and then 9 more.
Worked example 1.2.1
The temperature is -3 degrees. It rises by 7 degrees. What is the new temperature?

Method. Count up from -3 on a thermometer.

  1. Start at -3. The starting point is minus three.
    -3
  2. Count up 7 steps. Three steps go up to zero, and four more steps go on from zero.
    -3 -> 0 is 3 steps, then 4 more
  3. Write the new temperature. After 7 steps up from -3, we reach 4.
    4 degrees
  4. Write the answer.
    4

Check. -3 + 7 = 4, and 4 - 7 = -3. ✓

Worked example 1.2.2
Which is colder, -12 degrees or -3 degrees?

Method. The colder temperature is the smaller number.

  1. Compare the numbers. Both are negative, so the one further from zero is smaller.
    -12 is 12 below zero, -3 is 3 below zero
  2. Use the number line. -12 is further to the left.
    -12 < -3
  3. Write the answer. The colder temperature is the smaller number.
    -12
  4. Write the answer.
    -12

Check. -12 < -3, so -12 is colder. ✓

Worked example 1.2.3
Work out 9 - (-4).

Method. Subtracting a negative number is the same as adding the positive number.

  1. Count across zero. From -4 to 0 is 4 steps.
    4 steps
  2. Count on from zero. Then 9 more steps from 0 to 9.
    4 + 9
  3. Add the steps. The total distance is 13.
    9 - (-4) = 13
  4. Write the answer.
    13

Check. 13 + (-4) = 9. ✓

Watch out

Watch out. -8 is smaller than -2, even though 8 is bigger than 2. Look at the number line.

Exam tip

Practice guide. Draw a quick number line with zero in the middle when a question crosses zero.

Exercise 1B  Practice: negative numbers

Non-calculator
1
FThe temperature is -4 degrees. It rises by 6 degrees. What is the new temperature?
2
FThe temperature is 3 degrees. It falls by 8 degrees. What is the new temperature?
3
FWork out -2 - 5.
4
FWhat is the difference between -6 and 5?
5
FFill in the missing number: -8 + __ = 3
6
FWrite these numbers in order, from coldest to warmest: 2, -6, 0, -1.
7
PSThe temperature in a freezer is -18 degrees. It is raised by 25 degrees. What is the new temperature?
8
FTick the numbers that are less than -5. Options: -9 | -4 | -6 | 0 | -5. Write the options you would tick, in the order given.
9
FWhich is greater: -15 or -7? Write the greater number.
10
PSA submarine is at -120 metres. It rises 45 metres, then sinks 70 metres. What depth is it at now? Write the number.
11
PSWhich is the correct sign for the answer to: -4 - (-9)? Choose one: negative, positive or zero.
12
RWrite two numbers, one negative and one positive, whose difference is 10.
13
RIs it always true that a negative number is smaller than its positive partner? Explain using -5 and 5.
1.3

Order of operations and brackets

Year 6
Use the order of operations, including brackets (Year 6), Solve multi-step problems using the four operations (Year 5-6)
Video lesson: Order of operations and bracketsComing soon
Story. Mrs Okafor sells raffle tickets at the school fair. Adults pay 6 pounds and children pay 4 pounds. She writes a sum on the board to find the total, but first she must decide the order to work it out.

Rule

Work out brackets first. Then do multiplication and division (left to right). Then do addition and subtraction (left to right).

Brackets change the order. (2 + 3) x 4 = 20, but 2 + 3 x 4 = 14.

Key ideas

  • Brackets first, then x and divide, then add and subtract.
  • Work from left to right when the operations have the same priority.
  • A bracket can change the answer completely.
Worked example 1.3.1
Work out 12 + 8 x 3 - 4.

Method. Multiplication comes before addition and subtraction.

  1. Multiply first. 8 x 3 is done before adding.
    8 x 3 = 24
  2. Add and subtract from left to right. 12 + 24 = 36, then 36 - 4.
    12 + 24 - 4 = 32
  3. Write the answer.
    32

Check. 12 + 24 - 4 = 32. ✓

Worked example 1.3.2
Work out (12 + 8) x 3 - 4 / 2.

Method. Brackets first, then divide, then multiply and subtract.

  1. Work out the bracket. 12 + 8 = 20.
    (12 + 8) = 20
  2. Divide and multiply. Divide 4 by 2 and multiply 20 by 3.
    20 x 3 = 60 and 4 / 2 = 2
  3. Subtract. Take 2 from 60.
    60 - 2 = 58
  4. Write the answer.
    58

Check. (20 x 3) - 2 = 58. ✓

Worked example 1.3.3
Ticket sales: 4 adults pay 6 pounds each and 3 children pay 4 pounds each. Write one calculation for the total, then work it out.

Method. Work out each group, then add.

  1. Work out the adults. 4 x 6 = 24 pounds.
    24 pounds
  2. Work out the children. 3 x 4 = 12 pounds.
    12 pounds
  3. Add the two totals. Add the money from both groups.
    24 + 12 = 36 pounds
  4. Write the answer.
    36

Check. 4 x 6 + 3 x 4 = 36. ✓

Watch out

Watch out. 2 + 3 x 4 is 14, not 20. Multiply first, then add.

Exam tip

Practice guide. Underline the multiplication and division first, then do the sum from left to right.

Exercise 1C  Practice: order of operations

Non-calculator
1
FWork out 5 + 6 x 3.
2
FWork out 20 - 12 / 4.
3
FWork out (7 + 5) x 4.
4
FWork out 6 x (9 - 4) + 3.
5
FWork out 48 / (4 + 2) x 3 - 1.
6
PSWork out 100 - 3 x (2 + 8) + 5.
7
RPut one pair of brackets in 2 + 3 x 4 so that the answer is 20. Write the calculation with the brackets.
8
RWrite one pair of brackets in 10 - 4 / 2 + 1 so that the answer is 2. Write the calculation.
9
PSWork out 3 x 4 + 2 x 5 - 6 / 3.
10
PSA bag of 12 apples costs 3 pounds. Ravi buys 3 bags and pays 10 pounds. How much change does he get, in pounds?
11
PSWork out (25 - 7) x 5 / 6.
12
RTick the calculations that give an answer of 14. Options: 2 + 3 x 4 | (2 + 3) x 4 | 2 + 12 / 6 | 5 x 2 + 4 | 20 - 3 x 2. Write the options you would tick, in the order given.
1.4

Factors, multiples and primes

Year 6
Use factors, multiples and primes in problems (Year 5)
Video lesson: Factors, multiples and primesComing soon
Story. Nadia has 36 stickers and wants to put them into equal packs with none left over. She asks her friends how many packs she could make.

Rule

A factor divides exactly into a number. 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18, 36.

A multiple is in the times table of a number. 12 is a multiple of 3 and of 4.

A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are primes. 1 is not a prime.

Key ideas

  • Factors come in pairs: 4 x 9 = 36, so 4 and 9 are factors of 36.
  • The common factors of two numbers are the factors they share.
  • A prime number has exactly two factors.
Worked example 1.4.1
List all the factors of 36.

Method. Find pairs of numbers that multiply to 36.

  1. Start with 1. 1 x 36 = 36, so 1 and 36 are factors.
    1, 36
  2. Try 2, 3, 4, 5 and 6. 2 x 18, 3 x 12, 4 x 9 and 6 x 6 work. 5 does not divide 36.
    2, 18, 3, 12, 4, 9, 6
  3. Write them in order. Each number is listed once.
    1, 2, 3, 4, 6, 9, 12, 18, 36
  4. Write the answer.
    1, 2, 3, 4, 6, 9, 12, 18, 36

Check. Each factor divides 36 exactly. ✓

Worked example 1.4.2
What is the common factor of 12 and 18 that is largest?

Method. List the factors of each number, then find the largest one in both lists.

  1. List the factors of 12. 1, 2, 3, 4, 6, 12.
    1, 2, 3, 4, 6, 12
  2. List the factors of 18. 1, 2, 3, 6, 9, 18.
    1, 2, 3, 6, 9, 18
  3. Find the largest one in both lists. The biggest factor they share is 6.
    6
  4. Write the answer.
    6

Check. 12 / 6 = 2 and 18 / 6 = 3. ✓

Worked example 1.4.3
Which one of these numbers is prime: 39, 45, 47 or 51?

Method. A prime has only two factors, 1 and itself.

  1. Test 39. 39 = 3 x 13, so it is not prime.
    39 is not prime
  2. Test 45 and 51. 45 = 5 x 9 and 51 = 3 x 17, so neither is prime.
    45 and 51 are not prime
  3. Test 47. No number from 2 to 6 divides 47 exactly, so its only factors are 1 and 47.
    47 is prime
  4. Write the answer.
    47

Check. 47 has no factors other than 1 and 47. ✓

Watch out

Watch out. 1 is not a prime number, and 2 is the only even prime.

Exam tip

Practice guide. Check each candidate by dividing by the small primes, 2, 3, 5 and 7, before deciding it is prime.

Exercise 1D  Practice: factors, multiples and primes

Non-calculator
1
FList the factors of 24.
2
FWrite the first five multiples of 7.
3
FWhat is the highest common factor of 24 and 36?
4
FWhat is the lowest common multiple of 6 and 8?
5
FTick the prime numbers. Options: 9 | 11 | 15 | 17 | 21 | 29. Write the options you would tick, in the order given.
6
FWhich of these is a factor of 84? Write the number: 5, 6, 8.
7
PSWrite the prime factors of 60 in order, as a multiplication. Example format: 2 x 3 x 5
8
FWrite a prime number between 20 and 30.
9
PSTwo numbers multiply to make 42, and both are between 5 and 10. Write both numbers.
10
PSA school has 60 pupils and buses must carry the same number of pupils each. Each bus carries the same number of pupils. What is the greatest number of pupils that can go on each bus, with none left out?
11
RIs 91 prime? Choose the right answer. Choose one: yes, no, because 7 x 13 = 91 or no, because 2 x 45 = 91.
12
RTom says "Every multiple of 6 is also a multiple of 3." Is he right? Explain.
13
RIs the sum of two prime numbers always even? Give an example that proves your answer.
1.5

Choosing the operation and checking the answer

Year 6
Solve multi-step problems and decide which operation to use (Year 5-6), Use estimation to check whether an answer is sensible, and explain a method (Year 5-6)
Video lesson: Choosing the operation and checking the answerComing soon
Story. The school library has 1,840 books in 8 equal shelves. Mr Blake wants to know how many books are on each shelf, then he checks his answer by estimating.

Rule

Read the question twice. Decide which operations are needed and in which order.

Check the answer with an estimate: round each number first, work out a rough answer, and see if your answer is close.

Key ideas

  • Two steps: work out the total first, then share or take away.
  • Round to 1 significant figure to estimate, for example 368 x 5 is about 400 x 5 = 2000.
  • Always ask: is the answer about the right size?
Worked example 1.5.1
A bakery makes 368 rolls each day for 5 days. It sells 1,600 rolls. How many are left?

Method. Find the total first, then take away the sales.

  1. Find the total rolls made. 368 x 5 = 1,840 rolls.
    1,840 rolls
  2. Take away the rolls sold. Subtract the number sold from the total.
    1,840 - 1,600 = 240
  3. Check the answer. Rounding 368 up to 400 gives 400 x 5 = 2,000, and 2,000 - 1,600 = 400. The exact answer is smaller, because 368 was rounded up.
    about 400, and 240 is the exact answer
  4. Write the answer.
    240

Check. 368 x 5 = 1,840 and 1,840 - 1,600 = 240. ✓

Worked example 1.5.2
Estimate 4,980 x 6 by rounding, and say if the answer 29,880 is sensible.

Method. Round the first number, then multiply.

  1. Round 4,980 to the nearest 1,000. 4,980 is close to 5,000.
    5,000
  2. Estimate. 5,000 x 6 = 30,000.
    about 30,000
  3. Compare with the answer. 29,880 is very close to 30,000, so it is sensible.
    yes, 29,880 is sensible
  4. Write the answer.
    yes

Check. 5,000 x 6 = 30,000 and 30,000 - 29,880 = 120. ✓

Worked example 1.5.3
47 pupils go on a trip. Each bus seats 12 pupils. How many buses are needed?

Method. Divide, then decide whether to round up or down.

  1. Divide the pupils by the seats. 47 / 12 = 3 remainder 11.
    3 buses and 11 pupils
  2. Decide what the leftover means. The 11 pupils still need a seat, so one more bus is needed.
    3 + 1 = 4
  3. Write the answer. Four buses carry all 47 pupils.
    4
  4. Write the answer.
    4

Check. 4 x 12 = 48, which is enough for 47. ✓

Watch out

Watch out. When the answer is a remainder, read the question to decide whether to round up, down or ignore it.

Exam tip

Practice guide. Write the operation in words first: "total, then take away", or "share equally".

Exercise 1E  Practice: choosing the operation and checking

Non-calculator
1
FA pack has 36 pencils. A school buys 12 packs. How many pencils is that altogether?
2
FA cinema has 485 seats. 136 seats are empty. How many people are sitting down?
3
FEstimate 61 x 39 by rounding each number to the nearest 10. Write the estimate.
4
PSA bag of 8 apples costs 3 pounds 20 pence. Sana buys 4 bags. How much does she pay, in pounds?
5
PSTickets cost 7 pounds 50 pence each. Lena buys 5 tickets and pays with a 40 pound note. How much change does she get?
6
PSA tank holds 2,000 litres. It already has 1,375 litres. How many more litres can it hold?
7
PSMr Blake shares 1,840 books equally onto 8 shelves. How many books on each shelf?
8
RBen says "3,000 / 6 = 500, so 3,000 / 60 = 50." Is he right? Explain.
9
RSam works out 4,980 x 7 = 34,860. He checks it by working out 5,000 x 7 = 35,000 and says his answer is sensible. Is his check sensible? Choose one: Yes, 35,000 is close to 34,860, No, 4,980 x 7 must be 3,486 or No, 35,000 is much bigger than 34,860.
10
PSWrite a two-step word problem whose answer is 60, using the numbers 8 and 5 and the operations x and +.
11
PSRavi has 3 bags of 24 marbles and 14 loose marbles. He gives away 25 marbles. How many marbles does he have now?
12
PSA cake recipe needs 250 g of flour for 8 people. How much flour is needed for 24 people? Write the answer in grams.
13
RA school says "We need 600 chairs for 75 rows of 8 chairs." Is the number of chairs right? Explain with a calculation.

Chapter summary

SkillMethod
Place valueDigit x column value. 6 in 4,862,139 is worth 60,000.
RoundingLook at the digit next to the rounding column: 5 or more rounds up.
Negative numbersCount across zero on a number line; -8 is smaller than -2.
Order of operationsBrackets, then x and divide, then add and subtract.
Factors and multiplesFactors divide exactly; multiples are in the times table.
PrimesExactly two factors: 1 and itself. 1 is not prime.
Choosing the operationRead twice; decide total, take away, share or repeat.
CheckingRound and estimate; does the answer have the right size?

Can you…?

  • I can say the value of a digit in a number up to 10,000,000
  • I can round to the nearest 10, 100, 1,000 or 10,000 and explain my choice
  • I can count through zero and find differences between negative numbers
  • I can use brackets and the order of operations
  • I can choose the operation for a multi-step problem and check it with an estimate

Key terms

factor
a number that divides exactly into another number
multiple
a number in the times table of another number
prime number
a number with exactly two factors, 1 and itself
negative number
a number less than zero, such as -4
order of operations
the rule for which calculation to do first: brackets, then multiply and divide, then add and subtract
estimate
a rough answer found by rounding the numbers first

Exercise 1Z  Check-up

Non-calculator
1
What is the value of the digit 5 in 6,905,421?
2
Round 8,359 to the nearest 100.
3
Work out 9 x (6 + 4) - 15.
4
The temperature is -7 degrees. It rises by 12 degrees. What is the new temperature?
5
List the factors of 30 that are also multiples of 5 (write them all).
6
Put these numbers in order, from smallest to largest: 3,050, 30,500, 3,005, 35,000.
7
Work out 144 / 12 + 3 x 4.
8
A school has 642 pupils. 185 pupils are in Year 6. How many pupils are in the other years?
9
Which prime number is between 50 and 60?
10
Is it always true that a positive whole number is bigger than its negative partner? Explain with an example.
11
A bus company sells 2,450 tickets in January and 3,890 in February. How many tickets were sold in total?

Total 17 marks. Allow about 17 minutes.

Exercise 1X  Practice test questions

Non-calculator
1
FWhat is the value of the digit 5 in 3,507,946?[1 mark]
2
FRound 264,850 to the nearest 10,000.[1 mark]
3
FWrite the number 2,000,000 more than 4,375,210.[1 mark]
4
FThe temperature at 6 pm is -2 degrees. By midnight it has dropped by 5 degrees. What is the temperature at midnight?[1 mark]
5
PSWork out 5 + 20 / 4 x 3 - 1.[2 marks]
6
FWrite the lowest common multiple of 4 and 10.[2 marks]
7
PSA cinema has 24 rows of 36 seats. Only 700 seats are sold. How many seats are empty?[2 marks]
8
RExplain why 1 is not a prime number.[2 marks]
9
PSTick the numbers that are multiples of both 3 and 4. Options: 6 | 12 | 18 | 24 | 30. Write the options you would tick, in the order given.[2 marks]
10
FRound 6,249 to the nearest hundred.[1 mark]
11
RUse the digits 7, 3, 0 and 5 once each to make the smallest four-digit number. Write it.[2 marks]

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