Number Operations (CIE IGCSE Maths: Extended)

What kind of addition or subtraction questions could I be asked?

• Adding and subtracting could be a part of any question within your IGCSE course
  • Adding areas or volumes together in geometry questions
  • Working with estimated mean or other averages
  • Any problem solving questions in a variety of contexts
• A variety of vocabulary can be used to imply you must add or subtract values
  • For adding the word plus may be used or you could be asked to find the total or find the sum
  • For subtracting the word minus or take away may be used or you could be asked to find the difference
• Although you will have a calculator to carry out these sums on, you could be asked to show a non calculator method
  • You should be especially confident showing methods for adding and subtracting large numbers and decimals

How do I add large numbers?

• You will almost always be able to use your calculator in the exam, you only need to show a full column method if the question tells you to
  • If the question states without using your calculator then you must show a full written method, but you should still check your answer on your calculator 
• To add large numbers without a calculator
  • Write one number above the other in two rows, making sure that all the ones, tens, hundreds, and so on, are lined up in the same columns
  • Add the numbers in each column, writing the answer below the line
  • If your sum is a two digit number, split it into ones and tens, writing the value of the ones below the line, and the value of the tens at the top of the next column
  • If the sum of the last column is a 2 digit number, you can write the entire number below the line
    • this is effectively the same as writing it at the top of the next column, but it would be the only number in that column

How do I subtract large numbers?

• Again, use your calculator in the exam unless the question tells you not to
  • If the question states without using your calculator then you must show a full written method, but you should still check your answer on your calculator 
• To subtract large numbers without a calculator
  • Write one number above the other in two rows, making sure that all the ones, tens, hundreds, and so on, are lined up in the same columns
  • Subtract the bottom number in each column from the top number in each column, starting with the units column
    • If your subtraction produces a negative number, we can avoid this by “borrowing” a ten from the next column to the left
    • If you need to borrow from a column, but it is a zero, you can borrow a ten from the next column to the left; turning the 0 into a 10, which you can then borrow from
  • Repeat this for each of the next columns, until complete

How do I add or subtract with decimals?

• If the numbers involve decimals, make sure the decimal points are lined up in a column of their own and keep the decimal point in the answer in this column too
  • You may need to add zeros before or after the decimal point to keep everything in line
  • Again, check your answer with your calculator!

What kind of multiplication questions could I be asked?

• Multiplying two or more numbers could be a part of any question within your GCSE course, some examples are
  • Finding areas or volumes in geometry questions
  • Finding a distance given a speed and a time
  • Working with probabilities
  • Problem solving questions in a range of contexts
• A variety of vocabulary can be used to imply you must multiply two or more values
  • The word times or multiply may be used or you could be asked to find the product
• Although you will have a calculator to carry out these sums on, you could be asked to show a non calculator method
  • You should be especially confident showing methods for multiplying large numbers and decimals

How do I multiply two or more numbers without a calculator?

• It is unlikely you will have to do this without your calculator in the exam, however you should be confident with at least one method just in case
• Different methods work for different people, and some are better depending on the size of the numbers
• We recommend the following 3 methods depending on the size of number you are dealing with
  • If in doubt all methods will work for all numbers!

1. Lattice method

(Best for numbers with two or more digits)

• This method allows you to work with individual digits
• So in the number 3 516 you would only need to work with the digits 3, 5, 1 and 6 

So, 3516 × 23 = 80 868

Remember to check this with your calculator in the exam!

2. Partition method

• This method keeps the value of the larger number intact
• So with 3516 you would use 3000, 500, 10 and 6
• This method is not suitable for two larger numbers as you can end up with a lot of zero digits that are hard to keep track of

So, 3516 × 7 = 24 612

3. Repeated addition method

• This is best for smaller, simpler cases
• You may have seen this called ‘chunking’
• It is a way of building up to the answer using simple multiplication facts that can be worked out easily
  • eg. 13 × 23

1 × 23 = 23

2 × 23 = 46

4 × 23 = 92

8 × 23 =184

• • So, 13 × 23 = 1 × 23 + 4 × 23 + 8 × 23 = 23 + 92 + 184 = 299

How do I multiply with decimals?

• These 3 methods can easily be adapted for use with decimal numbers
• You ignore the decimal point whilst multiplying but put it back in the correct place in order to reach a final answer
• eg. 1.3 × 2.3
  • Ignoring the decimals this is 13 × 23, which from above is 299
  • There are two decimal places in total in the question, so there will be two decimal places in the answer
    • So, 1.3 × 2.3 = 2.99
• Remember to always use your calculator in the exam to check your answer

What kind of division questions could I be asked?

• Dividing two numbers could be a part of any question within your GCSE course, some examples are
  • Converting a fraction to a decimal
  • Finding a speed given a distance and a time
  • Finding a length or a width given an area
  • Problem solving questions in a range of contexts
• A variety of vocabulary can be used to imply you must divide one number by another
  • The word divide, quotient, share or per may be used
• Although you will have a calculator to carry out these sums on, you could be asked to show a non calculator method
  • You should be especially confident showing methods for dividing very small or large numbers and decimals

How do I divide a number by another without a calculator?

• It is unlikely you will have to do this without your calculator in the exam, however you should be confident with at least one method just in case
• Most students will have seen short division (bus stop method) and long division and there is often confusion between the two
• Fortunately, you only need one – so use short division
• While short division is best when dividing by a single digit, for bigger numbers you need a different approach
• You can use other areas of maths that you know to help – eg. cancelling fractions, “shortcuts” for dividing by 2 and 10, and the repeated addition (“chunking”) method covered in Multiplication

1. Short division (bus stop method)

• Apart from where you can use shortcuts such as dividing by 2 or by 5, this method is best used when dividing by a single digit

eg. 534 ÷ 6

So, 534 ÷ 6 = 89

2. Factoring & cancelling

• This involves treating division as you would if you were asked to cancel fractions
• You can use the fact that with division, most non-calculator questions will have only number answers
• The only thing to be aware of is that this might not be the case if you’ve been asked to write a fraction as a mixed number (but if you are asked to do that it should be obvious from the question)
  • eg. 1008 ÷ 28
    1008 ÷ 28 = 504 ÷ 14 = 252 ÷ 7 = 36
• You may have spotted the first two values (1008 and 28) are both divisible by 4 which is fine but if not, divide top and bottom by any number you can
• To do the last part (252 ÷ 7) you can use the short division method above

3. Dividing by 10, 100, 1000, … (Powers of 10)

• This is a case of moving digits along the place value columns

  eg.

  • 380 ÷ 10 = 38
    45 ÷ 100 = 0.45