Number Operations (CIE IGCSE Maths: Core)

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
  
  • The order in which numbers are added is not important
• The column method is the most commonly used hand written method
  • The numbers are written one number above the other,
    • Line up the digits using place value columns
  • Add each pair of corresponding digits from the top and bottom rows (work right to left)
  • If the result is a single digit
    • write the result in the relevant place value column below the line
  • If the result is a 2 digit number
    • the ones are written in the relevant place value column below the line
    • the tens are carried to the top of the next column
  • If the addition of the final pair of digits results in a 2 digit number 
    • write both digits below the line
• For example, the addition 9789 + 563 = 10 352
  

How do I subtract large numbers?

• A variety of written methods exist, but you only need to know one
  
  • The order in which two numbers are subtracted is important so ensure the calculation is the right way round
• The column method is the most commonly used hand written method
  • The numbers are written one number above the other
    • Line up the digits using place value columns
    • The number being subtracted should be below the original amount
  • Subtract each digit in the bottom value from the corresponding digit in the top value (work right to left)
  • If the digit being subtracted is bigger than the one it is subtracted from
    •  "borrow ten" from the next column to the left
• For example, 392 - 28 = 364
  

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
  • 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
  • Four common methods are described below, but there are many other valid methods

How do I use the column method?

• This is an efficient method if you are confident with multiplication
• To use the column method:
  • Write one number above the other lining up the digits using place value columns
  • Multiply the first digit (on the right) from the bottom value by each digit in the top value
    • Write the result under the line with the digits in the correct place value columns
  •  Multiply the next digit in the bottom value by each digit in the top value 
    • Always work from right to left
    • Use 0s as place holders when multiplying digits in columns other than the ones column
  • For example, 87 × 426 = 37 062

How do I use the lattice method?

• The lattice method is good for numbers with two or more digits
  • This method allows you to work with individual digits
• To use the lattice method:
  • Draw a grid
    • The number of rows should be the same as the number of digits in one number
    • The number of columns should be the same as the number of digits in the other number
    • Draw diagonal lines through the boxes
  • Multiply each pair of digits, writing the result in the relevant box
    • Ones should be written in the bottom half of the box and tens in the top half of the box
  • Add the digits along the diagonals and write the result in the diagonal outside the grid
    • Carry the tens of any 2 digit result into the next diagonal
• For example, 3516 × 23 = 80 868

How do I use the grid method?

• This method keeps the value of the larger number intact
  • It may take longer with two larger numbers
  • Be careful lining up numbers with lots of zeros!
• To use the grid method
  • Draw a grid
    • The number of rows should be the same as the number of digits in one number
    • The number of columns should be the same as the number of digits in the other number
  • Label the rows and columns with the values of each digit
    • E.g. For 3516 you would use 3000, 500, 10 and 6
  • Multiply together the relevant values and put the results in the boxes
  • Add up all of the cells in the boxes 
• For example, 3516 × 7 = 24 612

How do I use the repeated addition method?

• This is best for smaller, simpler cases
  • You may have seen this called ‘chunking’
• To use the repeated addition method
  • Build 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 4 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