2. 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.

3. 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.

4. 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 12 th one is the check digit

5. How is the Check Digit created? 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.

6. 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

7. Find the missing digit in each barcode 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?

8. (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

9. 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

10. Find the missing digit in each ISBN 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?.

11. 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?

12. 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…

13. Find the missing digit in each barcode Answers 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)

14. Find the missing digit in each ISBN Answers 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.

15. 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 I