Year 6 SATs Maths Revision: worked examples and questions with answers

Key words. The sum is the answer to an addition. A carry is the 1 ten (or 1 hundred) we move into the next column. The difference is the answer to a subtraction.

Method. Add the columns from the right, and carry any 1 ten into the next column.

1. Add the ones column. Add the two digits, and the carried number if there is one.
   8 + 5 = 13, so write 3 and carry 1
2. Add the tens column. Add the two digits, and the carried number if there is one.
   1 + 6 + 1 carried = 8, so write 8
3. Add the hundreds column. Add the two digits, and the carried number if there is one.
   7 + 2 = 9, so write 9
4. Add the thousands column. Add the two digits, and the carried number if there is one.
   4 + 3 = 7, so write 7
5. Write the answer. Read the column answers from left to right.
   7,983

Check. 7,983 − 3,265 = 4,718, so the sum is right. ✓

Watch out. Carry the 1 into the next column. Do not forget it, and do not write it in the same column.

Key words. The sum is the answer to an addition. A carry is the 1 ten (or 1 hundred) we move into the next column. The difference is the answer to a subtraction.

Method. Subtract the columns from the right. Exchange 1 from the next column when the top digit is too small.

1. Subtract the ones column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   4 is too small for 8, so exchange 1 ten: 14 − 8 = 6
2. Subtract the tens column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   0 - 1 (lent) gives -1, so exchange 1 ten: 9 − 7 = 2
3. Subtract the hundreds column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   0 - 1 (lent) gives -1, so exchange 1 ten: 9 − 6 = 3
4. Subtract the thousands column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   8 - 1 (lent) − 3 = 4
5. Write the answer. Read the answer from the bottom row.
   4,326

Check. 4,326 + 3,678 = 8,004, so the difference is right. ✓

Watch out. Exchange 1 from the next column, which is worth 10 of the column you are working in. Watch for zeros in the top number.

Key words. The sum is the answer to an addition. A carry is the 1 ten (or 1 hundred) we move into the next column. The difference is the answer to a subtraction.

Method. Subtract the columns from the right. Exchange 1 from the next column when the top digit is too small.

1. Subtract the ones column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   0 is too small for 7, so exchange 1 ten: 10 − 7 = 3
2. Subtract the tens column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   0 - 1 (lent) gives -1, so exchange 1 ten: 9 − 5 = 4
3. Subtract the hundreds column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   0 - 1 (lent) gives -1, so exchange 1 ten: 9 − 4 = 5
4. Subtract the thousands column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   6 - 1 (lent) − 2 = 3
5. Write the answer. Read the answer from the bottom row.
   3,543

Check. 3,543 + 2,457 = 6,000, so the difference is right. ✓

Watch out. Exchange 1 from the next column, which is worth 10 of the column you are working in. Watch for zeros in the top number.

Method. Use the column method, one digit at a time, and check with an estimate.

1. Multiply the ones digit. Multiply this digit, and add the carried number from the column before.
   7 × 6 = 42, so write 2 and carry 4
2. Multiply the tens digit. Multiply this digit, and add the carried number from the column before.
   4 × 6 + 4 carried = 28, so write 8 and carry 2
3. Write the carried number at the front. Nothing is left to multiply, so the carried number is written in front.
   2
4. Write the answer. This is the product.
   282

Check. Estimate: 50 × 6 = 300. The exact answer 282 is smaller than the estimate, because we rounded both numbers up. ✓

Watch out. Keep each column lined up. Multiply the tens row by 10 by writing a 0 first.

Method. Use the column method, one digit at a time, and check with an estimate.

1. Multiply the ones digit. Multiply this digit, and add the carried number from the column before.
   8 × 7 = 56, so write 6 and carry 5
2. Multiply the tens digit. Multiply this digit, and add the carried number from the column before.
   0 × 7 + 5 carried = 5, so write 5
3. Multiply the hundreds digit. Multiply this digit, and add the carried number from the column before.
   3 × 7 = 21, so write 1 and carry 2
4. Write the carried number at the front. Nothing is left to multiply, so the carried number is written in front.
   2
5. Write the answer. This is the product.
   2,156

Check. Estimate: 300 × 7 = 2,100. The exact answer 2,156 is bigger than the estimate, because we rounded both numbers down. ✓

Watch out. Keep each column lined up. Multiply the tens row by 10 by writing a 0 first.

Key words. In division, the dividend is the number being divided, the divisor is the number we divide by, the quotient is the answer, and the remainder is what is left over.

Method. Share the number into groups of the divisor, one digit at a time, and keep the remainder smaller than the divisor.

1. Divide 43 by 6. Use the first digits that make a number at least as big as 6.
   7 × 6 = 42; 43 − 42 = 1
2. Divide 12 by 6. Bring down the 2 to make 12.
   2 × 6 = 12; 12 − 12 = 0
3. Divide 6 by 6. Bring down the 6 to make 6.
   1 × 6 = 6; 6 − 6 = 0
4. Write the answer. The last remainder is what is left over, and it must be smaller than the divisor.
   721

Check. 721 × 6 + 0 = 4,326. ✓

Watch out. The remainder must be smaller than the number you divide by. If it is not, you can take one more group.

Key words. In division, the dividend is the number being divided, the divisor is the number we divide by, the quotient is the answer, and the remainder is what is left over.

Method. Share the number into groups of the divisor, one digit at a time, and keep the remainder smaller than the divisor.

1. Divide 35 by 4. Use the first digits that make a number at least as big as 4.
   8 × 4 = 32; 35 − 32 = 3
2. Divide 34 by 4. Bring down the 4 to make 34.
   8 × 4 = 32; 34 − 32 = 2
3. Divide 22 by 4. Bring down the 2 to make 22.
   5 × 4 = 20; 22 − 20 = 2
4. Write the answer. The last remainder is what is left over, and it must be smaller than the divisor.
   885 remainder 2

Check. 885 × 4 + 2 = 3,542. ✓

Watch out. The remainder must be smaller than the number you divide by. If it is not, you can take one more group.

Method. Use the column method, one digit at a time, and check with an estimate.

1. Multiply by the ones digit. Write 3 under the ones, and multiply the whole number by it.

      1,247
       × 53
   --------
      3,741
   + 62,350
   --------
     66,091

   1247 × 3 = 3741
2. Multiply by the tens digit, using a 0 place holder. The tens digit is worth ten times more, so write a 0 first.
   1247 × 5 = 6235, then 6235 × 10 = 62350
3. Add the two rows. Add the partial products to find the total.
   3741 + 62350 = 66,091
4. Write the answer. This is the product.
   66,091

Check. Estimate: 1,000 × 50 = 50,000. The exact answer 66,091 is bigger than the estimate, because we rounded both numbers down. ✓

Watch out. Keep each column lined up. Multiply the tens row by 10 by writing a 0 first.

Key words. In division, the dividend is the number being divided, the divisor is the number we divide by, the quotient is the answer, and the remainder is what is left over.

Method. Share the number into groups of the divisor, one digit at a time, and keep the remainder smaller than the divisor.

1. Divide 48 by 23. Use the first digits that make a number at least as big as 23.

         209
   23 ) 4816
        46
        --
         2
          0
          -
         21
         207
         ---
           9

   2 × 23 = 46; 48 − 46 = 2
2. Divide 21 by 23. Bring down the 1 to make 21.
   0 × 23 = 0; 21 − 0 = 21
3. Divide 216 by 23. Bring down the 6 to make 216.
   9 × 23 = 207; 216 − 207 = 9
4. Write the answer. The last remainder is what is left over, and it must be smaller than the divisor.
   209 remainder 9

Check. 209 × 23 + 9 = 4,816. ✓

Watch out. The remainder must be smaller than the number you divide by. If it is not, you can take one more group.

Key words. In division, the dividend is the number being divided, the divisor is the number we divide by, the quotient is the answer, and the remainder is what is left over.

Method. Share the number into groups of the divisor, one digit at a time, and keep the remainder smaller than the divisor.

1. Divide 62 by 35. Use the first digits that make a number at least as big as 35.

         177
   35 ) 6205
        35
        --
        27
        245
        ---
         25
         245
         ---
          10

   1 × 35 = 35; 62 − 35 = 27
2. Divide 270 by 35. Bring down the 0 to make 270.
   7 × 35 = 245; 270 − 245 = 25
3. Divide 255 by 35. Bring down the 5 to make 255.
   7 × 35 = 245; 255 − 245 = 10
4. Write the answer. The last remainder is what is left over, and it must be smaller than the divisor.
   177 remainder 10

Check. 177 × 35 + 10 = 6,205. ✓

Watch out. The remainder must be smaller than the number you divide by. If it is not, you can take one more group.

Key words. The order of operations says what to work out first: brackets, then multiplication and division from left to right, then addition and subtraction from left to right.

Method. Work out the multiplication and division first, then the addition and subtraction.

1. Work out the multiplication and the division first. Multiplication and division come before addition and subtraction.
   4 × 5 = 20 and 6 ÷ 2 = 3
2. Then add and subtract from left to right. Start at the left and work across.
   3 + 20 − 3 = 20
3. Write the answer.
   20

Check. Working strictly from left to right gives 3 + 4 = 7, 7 × 5 = 35, 35 - 6 = 29, and 29 ÷ 2 = 14.5, which is wrong. ✓

Watch out. Multiply and divide before you add and subtract.

Method. Use the column method, one digit at a time, and check with an estimate.

1. Multiply by the ones digit. Write 5 under the ones, and multiply the whole number by it.

      1,248
       × 25
   --------
      6,240
   + 24,960
   --------
     31,200

   1248 × 5 = 6240
2. Multiply by the tens digit, using a 0 place holder. The tens digit is worth ten times more, so write a 0 first.
   1248 × 2 = 2496, then 2496 × 10 = 24960
3. Add the two rows. Add the partial products to find the total.
   6240 + 24960 = 31,200
4. Write the answer. This is the product.
   31,200

Check. Estimate: 1,000 × 20 = 20,000. The exact answer 31,200 is bigger than the estimate, because we rounded both numbers down. ✓

Watch out. Keep each column lined up. Multiply the tens row by 10 by writing a 0 first.

Key words. The order of operations says what to work out first: brackets, then multiplication and division from left to right, then addition and subtraction from left to right.

Method. Work out the brackets first, then the multiplication and the division from left to right.

1. Work out the brackets first. Brackets come before every other operation.
   12 + 8 = 20 and 15 − 9 = 6
2. Multiply and divide from left to right. Here the multiplication comes first, then the division.
   20 × 6 = 120, then 120 ÷ 5 = 24
3. Write the answer.
   24

Check. Without the brackets the answer would be different: 12 + 8 × 15 - 9 ÷ 5 = 12 + 120 - 1.8, which is not 24. ✓

Watch out. Brackets change the order. Work them out first, even when the bracket is on the left.

Key words. The value of a digit is the digit times its place value. In 4,250, the 2 is in the hundreds place, so its value is 200.

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

1. Write the number in columns. Each digit sits in a column, with the ones on the right.
   2,450,000
2. Find the hundred thousands column. Count from the right: ones, tens, hundreds, and so on.
   2,450,000: the hundred thousands column has the digit 4
3. Multiply the digit by the place value. The value of a digit is the digit times the value of its column.
   4 × 100,000 = 400,000

Check. The other digits keep their own values, so this digit is worth 400,000 and no more. ✓

Watch out. The digit is worth its column value, not just the digit. A 6 in the thousands is worth 6,000.

Key words. The value of a digit is the digit times its place value. In 4,250, the 2 is in the hundreds place, so its value is 200.

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

1. Write the number in columns. Each digit sits in a column, with the ones on the right.
   47,305
2. Find the thousands column. Count from the right: ones, tens, hundreds, and so on.
   47,305: the thousands column has the digit 7
3. Multiply the digit by the place value. The value of a digit is the digit times the value of its column.
   7 × 1,000 = 7,000

Check. The other digits keep their own values, so this digit is worth 7,000 and no more. ✓

Watch out. The digit is worth its column value, not just the digit. A 6 in the thousands is worth 6,000.

Key words. To round, look at the digit just after the place you are rounding to. If it is 5 or more, round up. If it is less than 5, round down.

Method. Look at the digit after the hundreds place, then keep the digits up to that place and make the rest zeros.

1. Look at the digit after the hundreds place. That digit decides whether to round up or down.
   the digit is 5
2. Decide whether to round up or down. 5 is 5 or more, so round up.
   round up
3. Write the answer. Keep the digits before the hundreds place and write 0 in the places after it.
   6,900

Check. 6,900 is a multiple of 100, and it is the closest one to 6,850. ✓

Watch out. Do not just cut the digits off. Look at the digit after the hundreds place first.

Key words. To round, look at the digit just after the place you are rounding to. If it is 5 or more, round up. If it is less than 5, round down.

Method. Look at the digit after the ten thousands place, then keep the digits up to that place and make the rest zeros.

1. Look at the digit after the ten thousands place. That digit decides whether to round up or down.
   the digit is 4
2. Decide whether to round up or down. 4 is less than 5, so round down.
   round down
3. Write the answer. Keep the digits before the ten thousands place and write 0 in the places after it.
   250,000

Check. 250,000 is a multiple of 10,000, and it is the closest one to 254,689. ✓

Watch out. Do not just cut the digits off. Look at the digit after the ten thousands place first.

Key words. A negative number is less than zero. On a number line, numbers get smaller as you move left. The interval between two numbers is the number of steps from one to the other.

Method. Start with the most negative number, then go up the number line one number at a time.

1. Find the smallest number. Negative numbers are smaller than zero, and the bigger the number after the minus sign, the smaller the number.
   -10 is the smallest
2. Keep choosing the next smallest. Work along the list until every number has been chosen.
   -10, -4, 0, 3, 7
3. Write the answer. The numbers are now in order from smallest to biggest.
   -10, -4, 0, 3, 7

Check. Check on a number line: each number is to the right of the one before. ✓

Watch out. -10 is smaller than -4, even though 10 is bigger than 4.

Key words. A negative number is less than zero. On a number line, numbers get smaller as you move left. The interval between two numbers is the number of steps from one to the other.

Method. Start at -3 and move 8 steps down the number line, which is to the left.

1. Start at the first temperature. The first temperature is -3 degrees.
   -3 °C
2. Move down 8 degrees. Falling means going to the left, so take 8 away.
   -3 − 8 = -11
3. Write the answer. The unit is degrees Celsius, as in the question.
   -11

Check. Going back up 8 degrees from -11 gives -3 again. ✓

Watch out. A fall in temperature makes the number smaller, even when it is negative.

Key words. A negative number is less than zero. On a number line, numbers get smaller as you move left. The interval between two numbers is the number of steps from one to the other.

Method. Count the steps from the negative number up to zero, then from zero up to the positive number.

1. Count up from -12 to 0. Each step up the number line adds 1.
   12 steps
2. Count up from 0 to 5. Keep counting up, one step at a time.
   5 steps
3. Add the two parts of the interval. The whole interval is the steps to zero plus the steps from zero.
   12 + 5 = 17

Check. Going the other way: from 5 down to 0 is 5 steps, and from 0 down to -12 is 12 steps, so 17 altogether. ✓

Watch out. Do not subtract the two numbers. Crossing zero means you add the two distances.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the factors of both numbers, then pick the biggest one that is on both lists.

1. List the factors of the first number. A factor divides exactly into the number.
   factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
2. List the factors of the second number. Look for the factors that both lists share.
   factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
3. Find the biggest common factor. The highest common factor is the biggest number on both lists.
   common factors: 1, 2, 3, 4, 6, 12
4. Write the answer.
   12

Check. 24 ÷ 12 = 2 and 36 ÷ 12 = 3, both whole numbers. ✓

Watch out. Check that you have all the factors. Pairs of factors are easy to miss, so list them in order.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the multiples of each number, and find the smallest one that is on both lists.

1. List the multiples of 6. Multiply the number by 1, 2, 3 and so on.
   6, 12, 18, 24, 30, 36, 42, 48 ...
2. List the multiples of 8. Do the same for the other number.
   8, 16, 24, 32, 40, 48 ...
3. Find the smallest multiple on both lists. The first number that appears on both lists is the lowest common multiple.
   smallest common multiple = 24

Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3, so 24 is a multiple of both. ✓

Watch out. Do not stop at a number that is only on one list. Check both lists before you choose.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. A prime number has exactly two factors, 1 and itself. Test the primes up to the square root.

1. Look for factors up to the square root. The square root of 91 is about 9.5, so only small primes need testing.
   2, 3, 5, 7
2. Test each prime. Divide by each prime. If it goes in exactly, the number is not prime.
   2 does not divide 91; 3 does not divide 91; 5 does not divide 91; 91 ÷ 7 = 13
3. Write the answer. 7 divides into 91 exactly, so 91 has more than two factors.
   No

Check. The factors of 91 are 1, 7, 13, 91. ✓

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

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. A prime number has exactly two factors, 1 and itself. Test the primes up to the square root.

1. Look for factors up to the square root. The square root of 97 is about 9.8, so only small primes need testing.
   2, 3, 5, 7
2. Test each prime. Divide by each prime. If it goes in exactly, the number is not prime.
   2 does not divide 97; 3 does not divide 97; 5 does not divide 97; 7 does not divide 97
3. Write the answer. No prime up to 97 divides it, so 97 has only the factors 1 and 97.
   Yes

Check. The factors of 97 are 1, 97. ✓

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

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. Divide by the smallest prime that goes in, and keep going until you reach a prime. Write every prime as the answer.

1. Divide 60 by 2. Keep dividing by the smallest prime that goes in, until the answer is 1.
   60 ÷ 2 = 30
2. Divide 30 by 2. Keep dividing by the smallest prime that goes in, until the answer is 1.
   30 ÷ 2 = 15
3. Divide 15 by 3. Keep dividing by the smallest prime that goes in, until the answer is 1.
   15 ÷ 3 = 5
4. Divide 5 by 5. Keep dividing by the smallest prime that goes in, until the answer is 1.
   5 ÷ 5 = 1
5. Write the product. Write every prime you used, multiplied together.
   2 × 2 × 3 × 5

Check. 2 × 2 × 3 × 5 = 60. ✓

Watch out. Start with 2, then 3, then 5. Do not write a number that is not prime.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the factors of both numbers, then write down the ones that appear on both lists.

1. List the factors of the first number. Find every number that divides exactly into it.
   factors of 18: 1, 2, 3, 6, 9, 18
2. List the factors of the second number. Do the same for the second number.
   factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
3. Pick out the factors on both lists. A common factor is on both lists.
   common factors: 1, 2, 3, 6
4. Write the answer.
   1, 2, 3, 6

Check. The biggest common factor is 6, which is the highest common factor. ✓

Watch out. Do not forget 1, which is a factor of every number.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the factors of both numbers, then pick the biggest one that is on both lists.

1. List the factors of the first number. A factor divides exactly into the number.
   factors of 45: 1, 3, 5, 9, 15, 45
2. List the factors of the second number. Look for the factors that both lists share.
   factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
3. Find the biggest common factor. The highest common factor is the biggest number on both lists.
   common factors: 1, 3, 5, 15
4. Write the answer.
   15

Check. 45 ÷ 15 = 3 and 60 ÷ 15 = 4, both whole numbers. ✓

Watch out. Check that you have all the factors. Pairs of factors are easy to miss, so list them in order.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Divide the top and the bottom by the highest common factor.

1. Find the highest common factor of the top and bottom. The biggest number that divides into both is the one to divide by.
   factors of 18 and 24: highest common factor = 6
2. Divide the top and the bottom by 6. Dividing both by the same number keeps the fraction the same size.
   18 ÷ 6 = 3, and 24 ÷ 6 = 4
3. Write the answer. This fraction cannot be simplified any more.
   3/4

Check. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so the fraction keeps the same size. ✓

Watch out. Divide by the highest common factor, or by any common factor and then again, until nothing else divides.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Make the bottom numbers the same, then add or subtract the top numbers.

1. Find a common bottom number. The lowest common multiple of 4 and 6 is 12, so both fractions can use 12 as the bottom number.
   4 and 6: lowest common multiple = 12
2. Change both fractions to have the same bottom number. Multiply the top and bottom of each fraction by the same number.
   3/4 = 9/12, and 1/6 = 2/12
3. Add the top numbers. The bottom number stays the same.
   9 + 2 = 11, so 11/12
4. Simplify the answer. Divide the top and bottom by the same number to get the lowest terms.
   11/12 is already in its simplest form, so 11/12

Check. Both forms are the same amount: 11/12 is equal to 11/12. ✓

Watch out. Only add the top numbers. The bottom number stays the same when the fractions are added.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Make the bottom numbers the same, then add or subtract the top numbers.

1. Find a common bottom number. The lowest common multiple of 6 and 4 is 12, so both fractions can use 12 as the bottom number.
   6 and 4: lowest common multiple = 12
2. Change both fractions to have the same bottom number. Multiply the top and bottom of each fraction by the same number.
   5/6 = 10/12, and 1/4 = 3/12
3. Subtract the top numbers. The bottom number stays the same.
   10 − 3 = 7, so 7/12
4. Simplify the answer. Divide the top and bottom by the same number to get the lowest terms.
   7/12 is already in its simplest form, so 7/12

Check. Both forms are the same amount: 7/12 is equal to 7/12. ✓

Watch out. Only add the top numbers. The bottom number stays the same when the fractions are added.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Multiply the whole number by the bottom number, add the top number, then keep the same bottom number.

1. Multiply the whole number by the bottom number. Each whole has the same number of parts as the bottom number.
   3 × 5 = 15
2. Add the top number. Add the parts that are left over.
   15 + 2 = 17
3. Write it over the bottom number. The bottom number stays the same.
   17/5

Check. 17 ÷ 5 = 3 with 2 left over, so the answer is right. ✓

Watch out. Do not add the whole number to the top number. Multiply it by the bottom number first.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Divide the top number by the bottom number. The answer is the whole part, and the remainder goes over the bottom number.

1. Divide the top number by the bottom number. The whole number of times it goes in is the whole part.
   23 ÷ 4 = 5 remainder 3
2. Write the remainder over the bottom number. The remainder is the top number of the fraction part.
   the fraction part is 3/4
3. Write the answer.
   5 3/4

Check. 5 whole and 3/4 is 23/4, which is the same as 23/4. ✓

Watch out. The remainder goes on top of the fraction. The bottom number stays the same.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Divide by the bottom number to find one part, then multiply by the top number.

1. Find one part. Divide 96 into 8 equal parts.
   96 ÷ 8 = 12
2. Take 3 parts. The fraction is 3/8, so we need 3 of these parts.
   12 × 3 = 36

Check. 36 is 3 out of 8 equal parts of 96. ✓

Watch out. Divide by the bottom number first, then multiply by the top number.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Write both fractions with the same bottom number, then compare the top numbers.

1. Find a common bottom number. The lowest common multiple of 8 and 12 is 24.
   24
2. Change both fractions. Use the same bottom number, so only the top numbers need comparing.
   5/8 = 15/24, and 7/12 = 14/24
3. Compare the top numbers. The bigger top number is the bigger fraction.
   15 is bigger than 14
4. Write the answer.
   5/8

Check. Both fractions are now out of 24, so comparing 15 with 14 is a fair test. ✓

Watch out. A bigger bottom number does not mean a bigger fraction. Compare the fractions with the same bottom number.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. When you multiply by 10, 100 or 1000, every digit moves left. When you divide, every digit moves right.

Method. Line up the decimal points, add the numbers as whole numbers, then put the decimal point back.

1. Line up the decimal points. Write each number with 2 decimal places, so the digits in each column match.
   0.4 + 0.27
2. Work with the whole numbers. Ignore the points for now and work with whole numbers.
   40 + 27
3. Add the ones column. Add the two digits, and the carried number if there is one.
   0 + 7 = 7, so write 7
4. Add the tens column. Add the two digits, and the carried number if there is one.
   4 + 2 = 6, so write 6
5. Put the decimal point back. Put the point in the same place as the decimal points you lined up.
   0.67

Check. 0.67 − 0.27 = 0.4, so the sum is right. ✓

Watch out. Line up the decimal points first. Do not line up the last digits.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. When you multiply by 10, 100 or 1000, every digit moves left. When you divide, every digit moves right.

Method. Line up the decimal points, subtract the numbers as whole numbers, then put the decimal point back.

1. Line up the decimal points. Write each number with 2 decimal places, so the digits in each column match.
   3.6 − 1.85
2. Work with the whole numbers. Ignore the points for now and work with whole numbers.
   360 − 185
3. Subtract the ones column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   0 is too small for 5, so exchange 1 ten: 10 − 5 = 5
4. Subtract the tens column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   6 - 1 (lent) is too small for 8, so exchange 1 ten: 15 − 8 = 7
5. Subtract the hundreds column. Take the bottom digit from the top digit. If the top digit is too small, exchange 1 from the next column.
   3 - 1 (lent) − 1 = 1
6. Put the decimal point back. Put the point in the same place as the decimal points you lined up.
   1.75

Check. 1.75 + 1.85 = 3.6, so the difference is right. ✓

Watch out. Line up the decimal points first. Do not line up the last digits.

Key words. Per cent means out of 100. So 35 per cent means 35 out of 100, which is 0.35 as a decimal and 35/100 as a fraction.

Method. Find 1 per cent of the amount, then multiply it by the number of per cent.

1. Write the per cent and the amount. Per cent means out of 100.
   25 per cent of £40.00
2. Find 1 per cent. Per cent means out of 100, so divide the amount by 100.
   £40.00 ÷ 100 = £0.40
3. Multiply by 25. 25 per cent is 25 lots of 1 per cent.
   £0.40 × 25 = £10.00

Check. Check: £10.00 ÷ £40.00 = 0.25, which is 25 out of 100. ✓

Watch out. Divide by 100 first, then multiply. Do not multiply by 100.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Dividing by a whole number multiplies the bottom number, and keeps the top number.

1. Multiply the bottom number by the whole number. Sharing a fraction into 2 equal parts makes each part 2 times smaller, so the bottom number gets 2 times bigger.
   4 × 2 = 8
2. Keep the top number and simplify. The top number stays the same, then simplify if you can.
   3/8 = 3/8

Check. 3/8 × 2 = 3/4, so the division is right. ✓

Watch out. Multiply only the top number when you multiply by a whole number. Only the bottom number changes when you divide.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. When you multiply by 10, 100 or 1000, every digit moves left. When you divide, every digit moves right.

Method. Look at the digit after the place you are rounding to, then decide whether to round up.

1. Find the digit after the 2 decimal places. That digit decides whether to round up or down.
   4.678: the digit is 8
2. Decide. 8 is 5 or more, so round up.
   round up
3. Write the answer. Keep 2 decimal places.
   4.68

Check. 4.68 is the closest number to 4.678 with 2 decimal places. ✓

Watch out. Round the number, not the digits. Look only at the digit just after the place you are rounding to.

Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.

Method. Add the parts, share the total into one part, then multiply each share by its number of parts.

1. Add the parts of the ratio. The ratio 2:3 has 5 equal parts altogether.
   2 + 3 = 5
2. Find the value of one part. Share the total equally between all the parts.
   £60 ÷ 5 = £12
3. Multiply each share by its number of parts. The first share has 2 parts and the second has 3 parts.
   2 × £12 = £24, and 3 × £12 = £36
4. Write the answer.
   £24 and £36

Check. £24 + £36 = £60, so the shares add up to the total. ✓

Watch out. Share the total into all the parts, not just between two shares.

Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.

Method. Divide the known amount by its parts to find one part, then multiply one part by the other number of parts.

1. Find the value of one part. The first amount is 3 parts, so divide it by 3.
   18 ÷ 3 = 6
2. Find the second amount. The second amount is 5 parts, so multiply the value of one part by 5.
   5 × 6 = 30

Check. The two amounts keep the ratio 3:5, because 18 : 30 = 3:5. ✓

Watch out. Find one part first. Do not just multiply the known amount by the ratio.

Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.

Method. Find the cost of one item, then multiply by the number of items.

1. Find the cost of one item. 5 items cost £3.20, so share the cost equally.
   £3.20 ÷ 5 = £0.64
2. Multiply by 8. Each item costs the same, so multiply the cost of one item by 8.
   £0.64 × 8 = £5.12

Check. £5.12 ÷ 8 = £0.64, so the cost of one item is right. ✓

Watch out. Find the cost of one item first. Do not multiply the price of the first group by the number of items you want.

Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.

Method. Find the scale factor, then multiply the other length by it.

1. Find the scale factor. The scale factor is the new length divided by the old length.
   10 ÷ 4 = 2.5
2. Multiply the other side by the scale factor. Every length in the new shape is the old length times the same scale factor.
   6 × 2.5 = 15

Check. 15 ÷ 2.5 = 6, which is the old length. ✓

Watch out. Use the same scale factor for every length. Do not add the same amount to each side.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. Put the number in place of the letter, then multiply before you add.

1. Write the formula with n = 7. Put the number in place of the letter n.
   5n + 3 with n = 7
2. Multiply first. Multiplication comes before addition.
   5 × 7 = 35
3. Add the number. Add the last number to finish the calculation.
   35 + 3 = 38

Check. 38 − 3 = 35, and 35 ÷ 5 = 7, so the answer is right. ✓

Watch out. Multiply before you add: 3n + 4 with n = 5 is 15 + 4 = 19, not 35.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. Substitute the length and the width into the formula, working out the brackets first.

1. Write the formula. The perimeter of a rectangle is two lengths plus two widths.
   P = 2 × (l + w)
2. Add the length and the width. Work out the brackets first.
   8 + 5 = 13
3. Multiply by 2. There are two lengths and two widths.
   2 × 13 = 26

Check. 26 is the same as adding 8, 5, 8 and 5. ✓

Watch out. Work out the brackets first, then multiply by 2.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. Find how much the terms go up by, then find the number to add so that the first term is right.

1. Find the difference between terms. Subtract each term from the one after it.
   7, 12, 17, 22: each step adds 5
2. Write the part with n. The terms go up by 5 each time, so the rule has 5n.
   5n
3. Find the number to add. When n = 1 the term is 7, and 5 × 1 = 5, so add 2.
   5 × 1 + 2 = 7
4. Write the rule.
   5n + 2

Check. Test n = 2: 5 × 2 + 2 = 12, which is the second term. ✓

Watch out. Check the rule with the second term. Then the rule must work for every term.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. Put the number in place of the letter, then multiply before you add.

1. Write the formula with n = 10. Put the number in place of the letter n.
   4n + 3 with n = 10
2. Multiply first. Multiplication comes before addition.
   4 × 10 = 40
3. Add the number. Add the last number to finish the calculation.
   40 + 3 = 43

Check. 43 − 3 = 40, and 40 ÷ 4 = 10, so the answer is right. ✓

Watch out. Multiply before you add: 3n + 4 with n = 5 is 15 + 4 = 19, not 35.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. Undo the addition, then undo the multiplication. Do the same to both sides.

1. Take 7 away from both sides. Undo the addition first.
   3n = 31 − 7 = 24
2. Divide both sides by 3. Undo the multiplication to find n.
   n = 24 ÷ 3 = 8
3. Check the answer. Put n back into the first equation.
   3 × 8 + 7 = 31

Check. 3 × 8 + 7 = 31, which is the equation, so n = 8 is right. ✓

Watch out. Do the same thing to both sides. Undo the addition before the multiplication.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. Replace one letter so that there is only one unknown, then solve.

1. Write the two facts. One number is a, the other is b.
   a + b = 20 and a = 3b
2. Put the first fact into the second. Replace a with 3b in the sum.
   3b + b = 20, so 4b = 20
3. Divide by 4. Find the value of b.
   b = 20 ÷ 4 = 5
4. Find a. a is 3 times b.
   a = 3 × 5 = 15
5. Write the answer.
   15 and 5

Check. 15 + 5 = 20, and 15 is 3 times 5. ✓

Watch out. Replace a letter in one equation with the other. Then there is only one unknown.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. List every pair of whole numbers that adds to the total, and count each pair only once.

1. List the pairs, starting with the smallest number. Start with 1 and the biggest partner, and move towards the middle.
   1 + 11, 2 + 10, 3 + 9, 4 + 8, 5 + 7, 6 + 6
2. Count the pairs. Each pair is counted once, because order does not matter.
   6 pairs
3. Write the answer.
   6

Check. Every pair adds to 12, and no pair is missing from the list. ✓

Watch out. Count each pair once: 2 + 10 and 10 + 2 are the same pair.

Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.

Method. Work out the 10th term yourself and compare it with what Sam says.

1. Work out the term. Put n = 10 into the rule 3n + 2.
   3 × 10 + 2 = 30 + 2
2. Add. Add the number to the answer.
   30 + 2 = 32
3. Compare with Sam. Sam says the answer is 30, which is not 32.
   No, it is 32

Check. The first two terms are 5 and 8, which go up by 3 each time, so the rule is right. ✓

Watch out. Check Sam's claim by working out the term. Do not guess.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Use the number of m in one km, and multiply by it.

1. Write the fact about the units. There are 1000 m in 1 km.
   1 km = 1000 m
2. Multiply by 1000. To change km into m, multiply by 1000.
   3.4 × 1000 = 3400

Check. 3400 ÷ 1000 = 3.4, so the conversion is right. ✓

Watch out. Check the size: changing to a smaller unit gives a bigger number, and changing to a bigger unit gives a smaller number.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Use the number of g in one kg, and divide by it.

1. Write the fact about the units. There are 1000 g in 1 kg.
   1 kg = 1000 g
2. Divide by 1000. To change g into kg, divide by 1000.
   2500 ÷ 1000 = 2.5

Check. 2.5 × 1000 = 2500, so the conversion is right. ✓

Watch out. Check the size: changing to a smaller unit gives a bigger number, and changing to a bigger unit gives a smaller number.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Use the number of ml in one l, and multiply by it.

1. Write the fact about the units. There are 1000 ml in 1 l.
   1 l = 1000 ml
2. Multiply by 1000. To change l into ml, multiply by 1000.
   1.8 × 1000 = 1800

Check. 1800 ÷ 1000 = 1.8, so the conversion is right. ✓

Watch out. Check the size: changing to a smaller unit gives a bigger number, and changing to a bigger unit gives a smaller number.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Add up every side length, going round the shape once.

1. Write down every side. Go round the shape once, and write each side length.
   5 , 3 , 2 , 4 , 6 , 4
2. Add the sides. The perimeter is the total of every side.
   5 + 3 + 2 + 4 + 6 + 4 = 24

Check. Add the sides in a different order: the total is still 24. ✓

Watch out. Count every side, including the short ones.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Multiply the base by the perpendicular height.

1. Write the formula. The area of a parallelogram is the base times the perpendicular height.
   area = base × height
2. Multiply the base by the height. Use the height that is at right angles to the base, not the slanted side.
   12 × 7 = 84

Check. A parallelogram with the same base and height has the same area as a rectangle 12 by 7. ✓

Watch out. Do not use the slanted side. Use the perpendicular height.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Multiply the base by the height, then halve the answer.

1. Write the formula. A triangle is half of a rectangle with the same base and height.
   area = (base × height) ÷ 2
2. Multiply the base by the height. This gives the area of the rectangle.
   9 × 6 = 54
3. Halve it. The triangle is half the rectangle.
   54 ÷ 2 = 27

Check. Two of these triangles make a rectangle 9 by 6, with area 54. ✓

Watch out. Halve at the end. Do not forget the division by 2.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Multiply the length, width and height together.

1. Write the formula. The volume of a cuboid is length times width times height.
   volume = length × width × height
2. Multiply the first two numbers. Find the area of the base first.
   6 × 4 = 24
3. Multiply by the height. Each layer has this many cubes, and there are this many layers.
   24 × 5 = 120

Check. Count the layers: there are 5 layers of 24 cubes, which is 120. ✓

Watch out. The answer is in cubic units, such as cm³, not in square units.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Multiply the length, width and height together.

1. Write the formula. The volume of a cuboid is length times width times height.
   volume = length × width × height
2. Multiply the first two numbers. Find the area of the base first.
   5 × 3 = 15
3. Multiply by the height. Each layer has this many cubes, and there are this many layers.
   15 × 2 = 30

Check. Count the layers: there are 2 layers of 15 cubes, which is 30. ✓

Watch out. The answer is in cubic units, such as cm³, not in square units.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Add the hours and the minutes separately, and carry any 60 minutes into an hour.

1. Write the start time and the length of time. Use the start time, then add the time that passes.
   08:45 + 3 h 30 min
2. Add the hours. Add the whole hours first.
   8 + 3 = 11 hours
3. Add the minutes. If the minutes are 60 or more, take 60 away and add 1 more hour.
   45 + 30 = 75 minutes, which is 1 hour and 15 minutes
4. Write the finishing time. Give the time on the 24-hour clock with two digits for the hours and minutes.
   12:15

Check. Count on: 3 hours and 30 minutes after 08:45 reaches 12:15. ✓

Watch out. There are 60 minutes in an hour, not 100. Do not add the hours and minutes as one number.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Multiply the hours by 60, then add the extra minutes.

1. Change the hours to minutes. There are 60 minutes in 1 hour.
   2 × 60 = 120 minutes
2. Add the extra minutes. Add the minutes that are left over.
   120 + 35 = 155

Check. 155 ÷ 60 = 2 remainder 35, which is 2 hours and 35 minutes. ✓

Watch out. Multiply the hours by 60 first, then add the minutes.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Multiply the number of miles by 1.6, because 1 mile is 1.6 kilometres.

1. Write the rule. The question says 1 mile is 1.6 kilometres, so multiply the miles by 1.6.
   15 × 1.6
2. Multiply. Multiply the miles by 1.6 to get kilometres.
   15 × 1.6 = 24

Check. 24 ÷ 1.6 = 15, so the conversion is right. ✓

Watch out. Multiply to change miles into kilometres. A kilometre is a shorter unit, so the number gets bigger.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Work out the area and the perimeter of each shape, then compare them.

1. Work out the area of each shape. Multiply the length by the width.
   rectangle: 4 × 9 = 36 cm²; square: 6 × 6 = 36 cm²
2. Work out the perimeter of each shape. Add all four sides.
   rectangle: 4 + 9 + 4 + 9 = 26 cm; square: 6 + 6 + 6 + 6 = 24 cm
3. Compare. The areas are the same, but the perimeters are different.
   26 cm is not 24 cm

Check. Both shapes have an area of 36 cm², but the perimeters are 26 cm and 24 cm. ✓

Watch out. Two shapes with the same area can have different perimeters. Always work out both.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Use the fact about the total, add the angles you know, then take them away from the total.

1. Write the fact. Angles on a straight line add up to 180 degrees.
   the total is 180°
2. Add the angles you know. Add the angles that are given.
   67 = 67°
3. Take the total away. The missing angle a is what is left over.
   180 − 67 = 113

Check. 113 + 67 = 180, so the missing angle is right. ✓

Watch out. Use the total from the fact, not 90 or 360 by mistake.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Use the fact about the total, add the angles you know, then take them away from the total.

1. Write the fact. Angles at a point add up to 360 degrees.
   the total is 360°
2. Add the angles you know. Add the angles that are given.
   90 + 128 + 75 = 293°
3. Take the total away. The missing angle x is what is left over.
   360 − 293 = 67

Check. 67 + 293 = 360, so the missing angle is right. ✓

Watch out. Use the total from the fact, not 90 or 360 by mistake.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Use the fact about the total, add the angles you know, then take them away from the total.

1. Write the fact. Angles in a triangle add up to 180 degrees.
   the total is 180°
2. Add the angles you know. Add the angles that are given.
   48 + 67 = 115°
3. Take the total away. The missing angle x is what is left over.
   180 − 115 = 65

Check. 65 + 115 = 180, so the missing angle is right. ✓

Watch out. Use the total from the fact, not 90 or 360 by mistake.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Use the fact about the total, add the angles you know, then take them away from the total.

1. Write the fact. Angles in a quadrilateral add up to 360 degrees.
   the total is 360°
2. Add the angles you know. Add the angles that are given.
   110 + 75 + 80 = 265°
3. Take the total away. The missing angle x is what is left over.
   360 − 265 = 95

Check. 95 + 265 = 360, so the missing angle is right. ✓

Watch out. Use the total from the fact, not 90 or 360 by mistake.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Vertically opposite angles are equal.

1. Look at the two lines. The two lines cross at one point, making four angles.
   one angle is 58°
2. Find the angle opposite. Angles that are opposite each other when two lines cross are equal.
   opposite angle = 58°
3. Write the answer.
   58

Check. The angle next to it is 122 degrees, and 58 + 122 = 180, which is a straight line. ✓

Watch out. Opposite angles are equal. Angles next to each other add up to 180 degrees.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Double the radius to find the diameter.

1. Write the rule. The diameter goes right across the circle through the centre, so it is twice the radius.
   diameter = 2 × radius
2. Multiply the radius by 2. Double the radius to get the diameter.
   2 × 6.5 = 13

Check. The radius is half the diameter: 13 ÷ 2 = 6.5. ✓

Watch out. The radius is half the diameter. Do not halve the radius to find the diameter.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Find the total of the interior angles, then share it equally between the angles of the regular shape.

1. Write the number of sides. A regular shape has the same number of sides and angles.
   n = 6 sides
2. Find the total of the interior angles. A shape with 6 sides can be split into 4 triangles, and each triangle has 180 degrees.
   (6 − 2) × 180 = 720°
3. Share the total equally. A regular shape has 6 angles that are all the same.
   720 ÷ 6 = 120

Check. 120 × 6 = 720, so the answer is right. ✓

Watch out. Use the number of sides minus 2, not the number of sides, when you find the total.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Find the total of the interior angles, then share it equally between the angles of the regular shape.

1. Write the number of sides. A regular shape has the same number of sides and angles.
   n = 5 sides
2. Find the total of the interior angles. A shape with 5 sides can be split into 3 triangles, and each triangle has 180 degrees.
   (5 − 2) × 180 = 540°
3. Share the total equally. A regular shape has 5 angles that are all the same.
   540 ÷ 5 = 108

Check. 108 × 5 = 540, so the answer is right. ✓

Watch out. Use the number of sides minus 2, not the number of sides, when you find the total.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Multiply pi, which is 3.14, by the diameter.

1. Write the formula. The circumference is pi times the diameter. Use 3.14 for pi.
   C = 3.14 × 14
2. Multiply. Multiply 3.14 by the diameter.
   3.14 × 14 = 43.96

Check. The circumference is a little more than 3 times the diameter, and 43.96 is about 3 × 14. ✓

Watch out. Use the diameter, not the radius. If you are given the radius, double it first.

Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.

Method. Reflect the point across the axis: the distance from the axis stays the same, and the point is on the other side.

1. Write the point. Read the across number first, then the up number.
   (-3, 4)
2. Reflect the point. Reflecting in the y-axis changes the sign of the across number and keeps the up number.
   (3, 4)
3. Write the answer. The reflected point is the same distance from the mirror line on the other side.
   (3, 4)

Check. The reflected point is the same distance from the axis, on the opposite side. ✓

Watch out. Only one number changes sign. For the x-axis, the up number changes sign.

Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.

Method. Reflect the point across the axis: the distance from the axis stays the same, and the point is on the other side.

1. Write the point. Read the across number first, then the up number.
   (5, -2)
2. Reflect the point. Reflecting in the x-axis keeps the across number and changes the sign of the up number.
   (5, 2)
3. Write the answer. The reflected point is the same distance from the mirror line on the other side.
   (5, 2)

Check. The reflected point is the same distance from the axis, on the opposite side. ✓

Watch out. Only one number changes sign. For the x-axis, the up number changes sign.

Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.

Method. Move across by the first number, then up or down by the second number.

1. Move across. A positive number means right, a negative number means left.
   -2 + 4 = 2
2. Move up or down. A positive number means up, a negative number means down.
   3 − 5 = -2
3. Write the answer.
   (2, -2)

Check. Reversing both moves takes (2, -2) back to (-2, 3). ✓

Watch out. Move across first, then up or down. Keep the order of the numbers in the bracket.

Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.

Method. Find the x-value and the y-value that appear only once among the three corners.

1. Look at the x-values. Two corners share the same across number.
   x-values: 2, 6, 6
2. Look at the y-values. Two corners share the same up number.
   y-values: 2, 2, 6
3. Find the missing corner. The fourth corner uses the number that appears only once for x and for y.
   x = 2 and y = 6
4. Write the answer.
   (2, 6)

Check. The side from (2, 6) to each of the other corners is parallel to an axis. ✓

Watch out. Find the number that only appears once in each list. That is the missing corner.

Key words. A pie chart shows a whole split into sectors. A sector is one slice of the pie. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Divide the angle by 360, then take that fraction of the total.

1. Write the angle as a fraction of 360. The whole pie is 360 degrees.
   72 ÷ 360 = 1/5
2. Take that fraction of the total. The total is 150.
   1/5 × 150 = 30

Check. 30 out of 150 is the same fraction as 72 degrees out of 360. ✓

Watch out. Divide the angle by 360 first, not the total.

Key words. A pie chart shows a whole split into sectors. A sector is one slice of the pie. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Find the fraction of the whole pie, then take that fraction of 360 degrees.

1. Write the sector and the total. The sector is one part of the whole pie.
   Pasta: 18 out of 60
2. Write the sector as a fraction of the total. Pasta is 18 out of 60.
   18 ÷ 60 = 3/10
3. Multiply by 360 degrees. The whole pie is 360 degrees.
   3/10 × 360 = 108

Check. All the sectors must add up to 360 degrees. ✓

Watch out. Use 360 degrees for the whole pie, not 100.

Key words. A pie chart shows a whole split into sectors. A sector is one slice of the pie. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Write the sector as a fraction of the total, then multiply by 100.

1. Write the sector as a fraction of the total. Walk is 15 out of 60.
   15 ÷ 60 = 1/4
2. Change it to per cent. Per cent means out of 100.
   1/4 × 100 = 25

Check. 25 per cent of 60 is 15. ✓

Watch out. Per cent is out of 100.

Key words. A pie chart shows a whole split into sectors. A sector is one slice of the pie. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Change each per cent into an angle, then add the angles together.

1. Change each per cent to an angle. Each angle is the per cent of 360 degrees.
   35 per cent: 35 ÷ 100 × 360 = 126°; 25 per cent: 25 ÷ 100 × 360 = 90°; 20 per cent: 20 ÷ 100 × 360 = 72°; 20 per cent: 20 ÷ 100 × 360 = 72°
2. Add the angles. The angles of a pie chart must add up to 360 degrees.
   126 + 90 + 72 + 72 = 360

Check. 360 degrees is a whole turn, so the sectors fill the pie exactly. ✓

Watch out. Add the angles, not the per cents, when you check the whole pie.

Key words. A line graph marks a point for each time, and joins the points with straight lines. The scale shows the value of each point. The change is the end value minus the start value.

Method. Find the label, go up to the point, then read across to the scale.

1. Find Wed along the bottom. Start at the label.
   Wed
2. Go straight up to the point. Go up from the label to the point, and do not read the line between points.
   the point for Wed
3. Read across to the scale. Read the number on the scale level with the point.
   the sales on Wednesday = 50

Check. The point sits on a gridline, so the reading is exact. ✓

Watch out. Read the point, not the line between two points.

Key words. A line graph marks a point for each time, and joins the points with straight lines. The scale shows the value of each point. The change is the end value minus the start value.

Method. Subtract the start value from the end value.

1. Read the value at Mon. This is the start value.
   Mon = 20
2. Read the value at Fri. This is the end value.
   Fri = 60
3. Take the start value from the end value. The change is the end value minus the start value.
   60 − 20 = 40

Check. 20 + 40 = 60. ✓

Watch out. Start with the earlier point. Subtract the smaller value from the bigger value.

Key words. The mean is the average: add all the values, then share the total equally between the number of values. The total is all the values added together.

Method. Add the values, then divide the total by the number of values.

1. Add all the values. Count every value, including a zero.
   8 + 10 + 12 + 14 + 6 = 50
2. Count the values. Count how many numbers there are.
   there are 5 values
3. Divide the total by the number of values. Sharing the total equally gives the mean.
   50 ÷ 5 = 10

Check. 10 × 5 = 50, so the mean is right. ✓

Watch out. Divide by the number of values, not by the biggest value.

Key words. The mean is the average: add all the values, then share the total equally between the number of values. The total is all the values added together.

Method. Add the values, then divide the total by the number of values.

1. Add all the values. Count every value, including a zero.
   2 + 5 + 0 + 3 = 10
2. Count the values. Count how many numbers there are.
   there are 4 values
3. Divide the total by the number of values. Sharing the total equally gives the mean.
   10 ÷ 4 = 2.5

Check. 2.5 × 4 = 10, so the mean is right. ✓

Watch out. Divide by the number of values, not by the biggest value.

Key words. The mean is the average: add all the values, then share the total equally between the number of values. The total is all the values added together.

Method. Find the total from the mean, then take away the values you know.

1. Find the total from the mean. The total is the mean times the number of values.
   8 × 4 = 32
2. Add the values you know. These values are already given.
   6 + 9 + 11 = 26
3. Take the known total from the total. The missing value is what is left over.
   32 − 26 = 6

Check. 26 + 6 = 32, and 32 ÷ 4 = 8. ✓

Watch out. Find the total first. Do not subtract the mean from one value.

Key words. The mean is the average: add all the values, then share the total equally between the number of values. The total is all the values added together.

Method. Multiply the mean by the number of values to get the total.

1. Write the mean and the number of values. The mean is the total shared equally.
   mean 12, number of values 6
2. Multiply the mean by the number of values. Multiplying gives the total back.
   12 × 6 = 72

Check. 72 ÷ 6 = 12, so the total is right. ✓

Watch out. Multiply to find the total from the mean.

Key words. The mean is the average: add all the values, then share the total equally between the number of values. The total is all the values added together.

Method. Turn each mean into a total, add the totals, then divide by the number of values altogether.

1. Find the total of the first group. Multiply its mean by the number of values.
   12 × 5 = 60
2. Find the total of the second group. Do the same for the second group.
   15 × 5 = 75
3. Add the totals and count everyone. There are 5 + 5 = 10 values altogether.
   60 + 75 = 135, and 135 ÷ 10 = 13.5

Check. 13.5 × 10 = 135, so the combined mean is right. ✓

Watch out. Do not average the two means unless both groups have the same number of values.

Key words. The mean is the average: add all the values, then share the total equally between the number of values. The total is all the values added together.

Method. Work out the mean of a set of numbers, then check whether it is one of them.

1. Work out the mean. Add the values, then divide by how many there are.
   (1 + 2 + 6) ÷ 3 = 3
2. Compare the mean with the values. Look to see if the mean is one of the numbers in the list.
   3 is not one of 1, 2, 6
3. Write the answer. One example is enough to show that the mean is not always a value in the list.
   No

Check. Another set, such as 2, 4 and 6, has the mean 4, which is one of the values. So the answer depends on the numbers. ✓

Watch out. One example that does not match is enough to show that a statement is not always true.

Method. Use the column method, one digit at a time, and check with an estimate.

1. Multiply by the ones digit. Write 8 under the ones, and multiply the whole number by it.

       405
      × 28
   -------
     3,240
   + 8,100
   -------
    11,340

   405 × 8 = 3240
2. Multiply by the tens digit, using a 0 place holder. The tens digit is worth ten times more, so write a 0 first.
   405 × 2 = 810, then 810 × 10 = 8100
3. Add the two rows. Add the partial products to find the total.
   3240 + 8100 = 11,340
4. Write the answer. This is the product.
   11,340

Check. Estimate: 400 × 30 = 12,000. The exact answer 11,340 is close to the estimate. ✓

Watch out. Keep each column lined up. Multiply the tens row by 10 by writing a 0 first.

Key words. To round, look at the digit just after the place you are rounding to. If it is 5 or more, round up. If it is less than 5, round down.

Method. Look at the digit after the hundreds place, then keep the digits up to that place and make the rest zeros.

1. Look at the digit after the hundreds place. That digit decides whether to round up or down.
   the digit is 8
2. Decide whether to round up or down. 8 is 5 or more, so round up.
   round up
3. Write the answer. Keep the digits before the hundreds place and write 0 in the places after it.
   3,600

Check. 3,600 is a multiple of 100, and it is the closest one to 3,582. ✓

Watch out. Do not just cut the digits off. Look at the digit after the hundreds place first.

Key words. A factor divides exactly into a number. A multiple is in the times table of a number. A prime number has exactly two factors, 1 and itself. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share.

Method. List the multiples of each number, and find the smallest one that is on both lists.

1. List the multiples of 4. Multiply the number by 1, 2, 3 and so on.
   4, 8, 12, 16, 20, 24 ...
2. List the multiples of 6. Do the same for the other number.
   6, 12, 18, 24 ...
3. Find the smallest multiple on both lists. The first number that appears on both lists is the lowest common multiple.
   smallest common multiple = 12

Check. 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so 12 is a multiple of both. ✓

Watch out. Do not stop at a number that is only on one list. Check both lists before you choose.

Key words. A simplified fraction is in its lowest terms. A mixed number has a whole part and a fraction part. An improper fraction has a top number that is bigger than its bottom number.

Method. Make the bottom numbers the same, then add or subtract the top numbers.

1. Find a common bottom number. The lowest common multiple of 3 and 5 is 15, so both fractions can use 15 as the bottom number.
   3 and 5: lowest common multiple = 15
2. Change both fractions to have the same bottom number. Multiply the top and bottom of each fraction by the same number.
   2/3 = 10/15, and 1/5 = 3/15
3. Subtract the top numbers. The bottom number stays the same.
   10 − 3 = 7, so 7/15
4. Simplify the answer. Divide the top and bottom by the same number to get the lowest terms.
   7/15 is already in its simplest form, so 7/15

Check. Both forms are the same amount: 7/15 is equal to 7/15. ✓

Watch out. Only add the top numbers. The bottom number stays the same when the fractions are added.

Key words. Per cent means out of 100. So 35 per cent means 35 out of 100, which is 0.35 as a decimal and 35/100 as a fraction.

Method. Find 1 per cent of the amount, then multiply it by the number of per cent.

1. Write the per cent and the amount. Per cent means out of 100.
   20 per cent of £65.00
2. Find 1 per cent. Per cent means out of 100, so divide the amount by 100.
   £65.00 ÷ 100 = £0.65
3. Multiply by 20. 20 per cent is 20 lots of 1 per cent.
   £0.65 × 20 = £13.00

Check. Check: £13.00 ÷ £65.00 = 0.2, which is 20 out of 100. ✓

Watch out. Divide by 100 first, then multiply. Do not multiply by 100.

Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.

Method. Add the parts, share the total into one part, then multiply each share by its number of parts.

1. Add the parts of the ratio. The ratio 5:3 has 8 equal parts altogether.
   5 + 3 = 8
2. Find the value of one part. Share the total equally between all the parts.
   96 ÷ 8 = 12
3. Multiply each share by its number of parts. The first share has 5 parts and the second has 3 parts.
   5 × 12 = 60, and 3 × 12 = 36
4. Write the answer.
   60 and 36

Check. 60 + 36 = 96, so the shares add up to the total. ✓

Watch out. Share the total into all the parts, not just between two shares.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Multiply the length, width and height together.

1. Write the formula. The volume of a cuboid is length times width times height.
   volume = length × width × height
2. Multiply the first two numbers. Find the area of the base first.
   7 × 2 = 14
3. Multiply by the height. Each layer has this many cubes, and there are this many layers.
   14 × 3 = 42

Check. Count the layers: there are 3 layers of 14 cubes, which is 42. ✓

Watch out. The answer is in cubic units, such as cm³, not in square units.

Key words. The perimeter is the distance round the outside of a shape. The area is the space inside it, in square units. The volume is the space inside a solid, in cubic units.

Method. Add the hours and the minutes separately, and carry any 60 minutes into an hour.

1. Write the start time and the length of time. Use the start time, then add the time that passes.
   14:50 + 2 h 40 min
2. Add the hours. Add the whole hours first.
   14 + 2 = 16 hours
3. Add the minutes. If the minutes are 60 or more, take 60 away and add 1 more hour.
   50 + 40 = 90 minutes, which is 1 hour and 30 minutes
4. Write the finishing time. Give the time on the 24-hour clock with two digits for the hours and minutes.
   17:30

Check. Count on: 2 hours and 40 minutes after 14:50 reaches 17:30. ✓

Watch out. There are 60 minutes in an hour, not 100. Do not add the hours and minutes as one number.

Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.

Method. Find the total of the interior angles, then share it equally between the angles of the regular shape.

1. Write the number of sides. A regular shape has the same number of sides and angles.
   n = 10 sides
2. Find the total of the interior angles. A shape with 10 sides can be split into 8 triangles, and each triangle has 180 degrees.
   (10 − 2) × 180 = 1440°
3. Share the total equally. A regular shape has 10 angles that are all the same.
   1440 ÷ 10 = 144

Check. 144 × 10 = 1440, so the answer is right. ✓

Watch out. Use the number of sides minus 2, not the number of sides, when you find the total.

Key words. A pie chart shows a whole split into sectors. A sector is one slice of the pie. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Write the sector as a fraction of the total, then multiply by 100.

1. Write the sector as a fraction of the total. Bus is 24 out of 60.
   24 ÷ 60 = 2/5
2. Change it to per cent. Per cent means out of 100.
   2/5 × 100 = 40

Check. 40 per cent of 60 is 24. ✓

Watch out. Per cent is out of 100.

Key words. The sum is the answer to an addition. A carry is the 1 ten (or 1 hundred) we move into the next column. The difference is the answer to a subtraction.

Method. Add the columns from the right, and carry any 1 ten into the next column.

1. Add the ones column. Add the two digits, and the carried number if there is one.
   7 + 5 = 12, so write 2 and carry 1
2. Add the tens column. Add the two digits, and the carried number if there is one.
   0 + 4 + 1 carried = 5, so write 5
3. Add the hundreds column. Add the two digits, and the carried number if there is one.
   1 + 8 = 9, so write 9
4. Add the thousands column. Add the two digits, and the carried number if there is one.
   5 + 2 = 7, so write 7
5. Write the answer. Read the column answers from left to right.
   7,952

Check. 7,952 − 2,845 = 5,107, so the sum is right. ✓

Watch out. Carry the 1 into the next column. Do not forget it, and do not write it in the same column.

Key words. Per cent means out of 100. So 35 per cent means 35 out of 100, which is 0.35 as a decimal and 35/100 as a fraction.

Method. Divide the top number by the bottom number.

1. Write the division. A fraction line means divide the top by the bottom.
   7 ÷ 20
2. Do the division. Add a decimal point and zeros until the division ends.
   7 ÷ 20 = 0.35

Check. 0.35 × 20 = 7, so the decimal is right. ✓

Watch out. Keep adding zeros after the decimal point until the division ends.

Key words. The value of a digit is the digit times its place value. In 4,250, the 2 is in the hundreds place, so its value is 200.

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

1. Write the number in columns. Each digit sits in a column, with the ones on the right.
   1,070,482
2. Find the ten thousands column. Count from the right: ones, tens, hundreds, and so on.
   1,070,482: the ten thousands column has the digit 7
3. Multiply the digit by the place value. The value of a digit is the digit times the value of its column.
   7 × 10,000 = 70,000

Check. The other digits keep their own values, so this digit is worth 70,000 and no more. ✓

Watch out. The digit is worth its column value, not just the digit. A 6 in the thousands is worth 6,000.

Method. Use the column method, one digit at a time, and check with an estimate.

1. Multiply by the ones digit. Write 4 under the ones, and multiply the whole number by it.

      36
    × 24
   -----
     144
   + 720
   -----
     864

   36 × 4 = 144
2. Multiply by the tens digit, using a 0 place holder. The tens digit is worth ten times more, so write a 0 first.
   36 × 2 = 72, then 72 × 10 = 720
3. Add the two rows. Add the partial products to find the total.
   144 + 720 = 864
4. Write the answer. This is the product.
   864

Check. Estimate: 40 × 20 = 800. The exact answer 864 is close to the estimate. ✓

Watch out. Keep each column lined up. Multiply the tens row by 10 by writing a 0 first.

Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.

Method. Divide the known amount by its parts to find one part, then multiply one part by the other number of parts.

1. Find the value of one part. The second amount is 5 parts, so divide it by 5.
   35 ÷ 5 = 7
2. Find the first amount. The first amount is 2 parts, so multiply the value of one part by 2.
   2 × 7 = 14

Check. The two amounts keep the ratio 2:5, because 14 : 35 = 2:5. ✓

Watch out. Find one part first. Do not just multiply the known amount by the ratio.

Key words. A pie chart shows a whole split into sectors. A sector is one slice of the pie. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Find the fraction of the whole pie, then take that fraction of 360 degrees.

1. Write the sector and the total. The sector is one part of the whole pie.
   Dogs: 45 out of 60
2. Write the sector as a fraction of the total. Dogs is 45 out of 60.
   45 ÷ 60 = 3/4
3. Multiply by 360 degrees. The whole pie is 360 degrees.
   3/4 × 360 = 270

Check. All the sectors must add up to 360 degrees. ✓

Watch out. Use 360 degrees for the whole pie, not 100.