Number Operations (Edexcel GCSE Maths: Foundation)

What are number operations?

• Addition, subtraction, multiplication and division are the four basic number operations
• For the non-calculator paper, a written method will be required
• Different vocabulary may be used for each of the operations
  • Addition may be phrased using plus, total or sum
  • Subtraction may be phrased using difference or take away
  • Multiplication may be phrased using lots of, times or product
  • Division may be phrased using quotient, share and per

How do I add large numbers without a calculator?

• There are a variety of written methods that can be used to add large numbers
  • You only need to know one method, but be able to use it confidently
  • The method described below is commonly known as the column method
  • The order in which numbers are added is not important
• STEP 1
  Write one number above the other in two rows, making sure that the place values are lined up in the same columns
  e.g.  9789 + 563 would be written as   
• STEP 2
  Add the digits in the ones (units) column
  Write the answer below the line in the ones column
  However, if your sum is a two digit number, split it into ones and tens
  Write the value of the ones below the line in the current column, and the value of the tens at the top of the next (to the left) column
  e.g. for the units column, 9+3=12
     
• STEP 3
  Repeat STEP 2 for each column, in increasing place value order (i.e. moving a place to the left each time)
  e.g.  For the tens column, 1+8+6=15
     
• If the sum of the last (highest place value) column is a two digit number, you can write the entire number below the line
  • This is effectively the same as writing the tens digit at the top of the next column, but it would be the only digit in that column
• The final answer for this example would look like this:
  

How do I subtract large numbers?

• A variety of written methods exist, but you only need to know one
  • The method described below is the (subtraction) column method
  • The order in which two numbers are subtracted is important so ensure the calculation is the right way round
• STEP 1
  Write the number that is being subtracted below the original amount, making sure that the place values are lined up in the same columns
  e.g.  392 - 28 would be written as   
• STEP 2
  In the ones (units) column, subtract the bottom number from the top number, writing the answer below the answer line
  If the top number is smaller than the bottom number then "borrow ten" from the next column (to the left) then continue as above
  e.g.   
• If the column being borrowed from is a zero, "borrow ten" from the next column again, turning the zero into a ten, which can then be borrowed from
• STEP 3
  Repeat STEP 2 for each column, in increasing place value order (i.e. moving a place to the left each time)
  e.g.     
• If you find you need to borrow in the last (highest place value) column, an error in one of the previous steps may have been made, so check your working carefully

How do I add or subtract with decimals?

• If the numbers involve decimals, the column methods for addition and subtraction are the same as above
  • In STEP 1, line the place value columns up carefully, and make sure the decimal points are all in the same column
    
    •  Writing zeros (often called place value holders) can help keep everything in line
  • e.g.  2.145 + 13.02 would be written as   

How do I multiply two numbers without a calculator?

• There are a variety of written methods that can be used to add large numbers
  • You only need to know one method, but be able to use it confidently
  • Three common methods are described below, but there are many other valid methods

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 
• To find 3516 × 23 :

So, 3516 × 23 = 80 868

2. Grid method

• This method keeps the value of the larger number intact
• So with 3516 you would use 3000, 500, 10 and 6
• This method may take longer with two larger numbers as you can end up with many terms to add up
  • Be wary of large numbers with lots of zeros; line up the columns carefully in the final step
• To find 3516 × 7 :

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
  • To find 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 decimals without a calculator?

• These 3 methods can be easily adapted for use with decimal numbers
• Ignore the decimal point whilst multiplying but put it back in the correct place for the final answer
• eg. 1.3 × 2.3
  • Ignoring the decimals this is 13 × 23, which can be worked out as 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
    • Be careful that 1.30 is the same as 1.3, so 1.30 is still only 1 decimal place
  • Estimating is a good check too
    • 1.3 × 2.3 is approximately 1.5 × 2, which is 3
    • So we know the answer should be close to 3, rather than 0.3 or 30

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

• The most common written method for division is short division (or "the bus stop method")
  • There are other methods such as long division, but short division is generally the most efficient
• While short division is best when dividing by a single digit, for bigger numbers you may need a different approach
• You can use other number skills to help
  • eg. cancelling fractions, “shortcuts” for dividing by 2 and 10, and the repeated addition (“chunking”) method covered in Multiplication

Short division (bus stop method)

• Unless you can use simple shortcuts such as dividing by 2 or by 10, this method is best used when dividing by a single digit

To find 174 ÷ 3

• Starting from the left; 3 fits into 1, 0 times, with a remainder of 1
  • Carry the remainder of 1 over to the next digit, which forms 17
  • 
• 3 fits into 17, 5 times, with a remainder of 2
  • Carry the remainder of 2 over to the next digit, which forms 24
  • 
• 3 fits into 24, 8 times, with no remainder
  • 
• So, 174 ÷ 3 = 58

Factoring & cancelling

• This involves treating division as you would if you were asked to simplify fractions
  
  • For example, 1008 ÷ 28 can be written as
  • This can then be simplified
    • 
    • 252 ÷ 7 can then be calculated using short division; the answer is 36

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

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

  For example

  • 380 ÷ 10 = 38.0 (shifts by 1 column)

  • 45 ÷ 100 = 0.45 (shifts by 2 columns)

  • 28 ÷ 1000 = 0.028 (shifts by 3 columns)
    • For cases like this, it can help to add leading zeros
    • 0028 ÷ 1000 may be easier to visualise

How do I divide a decimal without a calculator?

• Use a similar approach to when multiplying decimals
• Ignore the decimal point whilst carrying out the division, but put it back in the correct place for the final answer
• e.g.  4.68 ÷ 6
  • Ignoring the decimal, this is 468 ÷ 6, which can be worked out as 78
  • There are two decimal places in total in the question, so there will be two decimal places in the answer
    • So, 4.68 ÷ 6 = 0.78
    • Be careful if 6 is written as 6.0 instead, this is still zero decimal places
  • Estimating is a good check too
    • 4.68 ÷ 6 is approximately 5 ÷ 6, which is between 1 and 0.5
    • So we know the answer should be 0.78, rather than 7.8 or 0.078