Transcript here - MEI

Barcodes and ISBN numbers:
which are better at detecting errors?
• 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.
Check digits
• 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.
Barcodes
• 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 12th one is the check digit
How is the Check Digit created?
• Find the sum of the 1st, 3rd, 5th, etc…
• Find the sum of the 2nd, 4th, 6th, 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.
Example
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
Find the missing digit in each barcode
•
•
•
•
•
14373582194?
25632852526?
?58253481077
3 6 ?1 2 8 5 3 2 2 7 6
4?7239128321
In each case, is there only one possibility?
Can you find examples where there are several
alternatives for the missing digit?
(Old) ISBNs
• 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
Example
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
Find the missing digit in each ISBN
• 014352146?
• 021364525?
• 02152386?1
• 0?1325475X
• 02035?3215
In each case, is there only one possibility?
Can you find examples where there are several
alternatives for the missing digit?
Which is most reliable?
• 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?
Teacher Notes
• 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…
Find the missing digit in each barcode
Answers
• 143735821949
• 256328525264
• 458253481077
• 365128532276
• 427239128321
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)
Find the missing digit in each ISBN
Answers
•
•
•
•
•
0143521462
0213645254
0215238621
011325475X
0203513215
The missing numbers are always unique
Encourage pupils to think about why this is.
Exploration ‘answers’
• 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