• Virtually all packaged products have a barcode on so that optical readers can recognise the item.
• ISBNs (International Standard Book Numbers) have been in existence since 1970 and until
2007 had 10 digits.
• Since 2007, ISBNs have changed to a 13 digit format.

• Both barcodes and ISBNs have a ‘check digit’ which alerts users to mistakes which may have occurred in writing or typing the number. These are created in two different ways
• A key question is how many mistakes does each pick up? Essentially, which is best?
• To be able to explore this, we need to understand how check digits are created in both types of code.

• There are several different lengths of barcode, but 12 and 13 digit ones are the most common.
• Looking at a 12 digit barcode on an item, the first
11 digits represent the number for the item and the 12 th one is the check digit

• Find the sum of the 1 st , 3 rd , 5 th , etc…
• Find the sum of the 2 nd , 4 th , 6 th, etc… and then multiply it by 3
• The two subtotals are then added together
• The check digit (0 to 9) is the number that should be added to the total to make the next multiple of 10.

For an item number of
8 1 3 4 2 6 3 7 2 0 4
8 + 3 + 2 + 3 + 2 + 4 = 22
(1 + 4 + 6 + 7 + 0) x 3 = 54
54+22 = 76 therefore the check digit is 4

•
1 4 3 7 3 5 8 2 1 9 4 ?
• 2 5 6 3 2 8 5 2 5 2 6 ?
• ?
5 8 2 5 3 4 8 1 0 7 7
• 3 6 ?
1 2 8 5 3 2 2 7 6
• 4 ?
7 2 3 9 1 2 8 3 2 1
In each case, is there only one possibility?
Can you find examples where there are several alternatives for the missing digit?

• Each ISBN is a 10 digit number, the tenth one being the check digit.
• To obtain the check digit, each digit is multiplied by a different number (from 10 descending by 1 each time)
• The check digit makes the sum of the totals up to a multiple of 11

For a book number of:
0 2 5 4 2 6 3 4 2
(10x0)+(9x2)+(8x5)+(7x4)+(6x2)+(5x6)+(4x3)+(3x4)+(2x2) = 156
Multiples of 11:
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176
So after 156, the next multiple of 11 is 165, which means the check digit is 9
Note: if a check ‘digit of 10 is required, an X is used

• 0 1 4 3 5 2 1 4 6 ?
• 0 2 1 3 6 4 5 2 5 ?
• 0 2 1 5 2 3 8 6 ?
1
• 0 ?
1 3 2 5 4 7 5 X
• 0 2 0 3 5 ?
3 2 1 5
In each case, is there only one possibility?
Can you find examples where there are several alternatives for the missing digit?

• Mistakes can be made when writing down or typing out long numbers – which is why the check digit is used
• Transcription errors are simply when a single wrong digit is used
• Transposition errors are where two (or more) neighbouring digits appear in the wrong order
• Explore how good each of the checking mechanisms are in picking up each of these errors
• Can you find an error that won’t be picked up?

• This material is accessible to most Key Stage 3 and 4 pupils
• The initial part of the lesson focuses on pupils understanding how check digits are created and the mathematical content involved is simple arithmetic
• The later part of the lesson asks pupils to explore errors.
This will require them to use a range of problem-solving and strategy skills as well as developing a sense of number.
• Teachers might like to add their own scaffolding to this part of the lesson for some or all pupils
• Pupils can debate which system is most reliable based on their findings…

• 1 4 3 7 3 5 8 2 1 9 4 9
• 2 5 6 3 2 8 5 2 5 2 6 4
• 4 5 8 2 5 3 4 8 1 0 7 7
• 3 6 5 1 2 8 5 3 2 2 7 6
• 4 2 7 2 3 9 1 2 8 3 2 1
The missing numbers are always unique
Encourage pupils to think about why this is.
(the end digit for multiples of 3 are unique from 0x3 to 9x3)

• 0 1 4 3 5 2 1 4 6 2
• 0 2 1 3 6 4 5 2 5 4
• 0 2 1 5 2 3 8 6 2 1
• 0 1 1 3 2 5 4 7 5 X
• 0 2 0 3 5 1 3 2 1 5
The missing numbers are always unique
Encourage pupils to think about why this is.

• Both systems will detect many errors.
• A common error is a simple transposition of two neighbouring digits. In barcodes this is usually detected, in ISBNs it is always detected
• There are a number of errors that will not be detected. e.g. Barcodes: transposing any two digits in ‘next but one’ positions such that
1 4 3 7 3 5 8 2 1 9 4 9 becomes 1 4 3 5 3 7 8 2 1 9 4 9
However, with ISBNs this type of error will be detected
(though it is perhaps a strange error to make!)
• With both systems ‘random errors’ will sometimes be detected, and sometimes not
I