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