Prior to the lesson, collect samples of barcodes, ISBN, UPC, and sample
credit cards (from mail advertisements or from local financial
institutions). Not only will these examples provide students with a
visual, but they can also be used as part of the lesson when verifying
the accuracy of the number. Begin by providing a background for the
creation of barcoding systems and how these systems are used today.
The first patent for barcodes was issued to Bernard Silver and
Norman Woodland in 1952. Since then these coding systems have expanded,
been modified, and applied to a variety of areas. One of the most
common uses is in retail and grocery stores. Although the barcode is
not the price of the item, it does allow for the item to be registered
with an associated price. When the bar code is scanned, the associated
price will be read by the cash register. Other uses are found in
monitoring blood supplies, identification on prescription drugs, book
checkout at libraries, tracking luggage, and express shipping services.
Another advantage to the barcode system is that when an additional digit is included, known as the check digit,
many errors that occur during data entry can be detected. These errors
occur when passing information over the phone or internet. It is quite
easy for people to transpose numbers (45 when it should be 54), replace
a single digit with another, omit digits, or double an incorrect digit
(799 is entered as 779). Using a check digit within a check equation
helps to catch these errors and verify the validity of the number. As
students will see in the ISBN barcode, the number is also used for
Begin by taking samples of barcodes that you have collected from items
that are sold in stores. This type of barcode is referred to as the
Universal Product Code, or UPC. Students will be instructed as to the
algorithm and then determine the validity of the number. Next, students
should be challenged to determine a check digit for a barcode. The UPC
system uses a mod 10 congruence. This system uses a weighting factor of
3 for the digits in the even positions. This means that even-positioned
digits will be multiplied by three.
For the first example, use the given UPC symbol 7-86936-24425-0 from the movie "The Incredibles."
To verify this number, follow the steps:
- Every even-positioned digit, counting from the right to left, will
be multiplied by 3. All odd-positioned digits will be multiplied by 1.
1(0) + 3(5) + 1(2) + 3(4) + 1(4) + 3(2) + 1(6) + 3(3) + 1(9) + 3(6) + 1(8) + 3(7)
- Sum the products.
0 + 15 + 2 + 12 + 4 + 6 + 6 + 9 + 9 + 18 + 8 + 21 = 110
- Determine the validity by dividing the sum by 10.
110 ÷ 10 = 11 remainder 0. Therefore 110 mod 10 = 0.
This is a valid UPC number.
Students may decide to apply the distributive property and multiply 3
by the sum of the even digits and multiply 1 by the sum of the odd
Next provide students with the UPC number 7-96714-78601-y, where y
is the check digit. Using the process from above, students should
determine the check digit. The sum of the products is 112. The check
digit must be 8, because (112+8) mod 10 = 0. On Question 1 of the Check That Digit activity sheet, students will verify the check digit for two UPC numbers.
Another barcode system is the International Standard Book Number, or
ISBN. This system was developed in the late 1960’s and early 1970’s. It
became apparent that there needed to be a uniform system that would
identify books that were published throughout the world. Now every book
could have a special identification number. The ISBN is a ten‑digit
number composed of blocks of numbers that have different meaning. There
are four parts to the number, which are separated by hyphens or spaces.
The first part of the number identifies the language or country
(referred to as the group identifier) and is at most five digits. The
second part of the number identifies the publisher and may be at most
seven digits. The third part of the number represents the item number
or edition for that publisher. It may consist of no more than six
digits. The final part is the check digit. Part of the flexibility of
this system is the fact that there are many numbers available to be
used. Recall that there are a maximum of 10 digits with the 10th being
reserved for the check digit. Therefore, the first three parts of the
number must have a combined total of nine digits. Leading zeroes are
used as place fillers in the event there would not be enough digits in
a particular section to ensure there are an appropriate number of
digits. The diagram below shows an example of an ISBN number.
The check digit is calculated differently than that of the UPC
system. Begin by multiplying the first digit by 10, the second by 9,
the third by 8, and continue in this fashion until the ninth digit is
multiplied by 2. Next, determine the sum of these products. This is a
modulus‑11 system, which means that the sum of the products of the
first nine digits plus the check digit must be a multiple of 11. One
problem that arises in this process is that the check digit might need
to be a 10. Because we only have digits 0‑9, an X is written in the
check‑digit place. (The X is reflective of the Roman numeral for 10.)
Questions 3‑6 on the Check That Digit
activity sheet deal specifically with the ISBN. Note: Beginning January
1, 2007, the current ISBN system will be replaced with the ISBN-13
system (for further information, see the ISO Web Site).
It is a 13-digit number beginning with 978, followed by the current
nine digits of the ISBN, and it will have a new check digit. The check
digit will be found using a method different from the current one. When
all old ISBN’s have been used, the next series will begin with 979.
Credit cards use a system of blocked numbers similar to the ISBN. One
obvious difference is that the maximum length for the number is
19 digits, although many numbers range from 13‑16 digits.
The first digit of a credit card number is the Major Industry
Identifier (MII) and identifies which group issued the card, as shown
For instance, a number beginning with a 3 would be
representative of the travel and entertainment category. The American
Express card falls into this category. Cards issued by gas companies
are given the beginning digit 7. The popular Visa and MasterCard fall
under the banking and financial category (4, 5). The next block of
numbers is the Issue Identifier. Including the MII digit, the Issue
Identifier is six digits long. The account number begins with the
seventh digit and ends with the next‑to‑last digit. The final digit is
the check digit.
The process used to determine the check digit is the Luhn algorithm
(mod 10), named after IBM scientist Hans Peter Luhn. This algorithm
works as follows:
- Begin by doubling all even-positioned digits when counting from
right to left. If doubling results in a two‑digit number, add the
digits. For instance, if the original digit were a 6, doubling it would
give 12, so use 1 + 2 = 3.
- Determine the sum of the results from Step 1 and each of the unaffected (odd‑positioned) digits in the original number.
- Verify the account number by determining if the sum from Step 2 is a multiple of 10.
Before proceeding to the questions on the activity sheet pertaining
to this topic, have students become more familiar with the Luhn
algorithm by determining the validity of the check digit for the
following account number: 5314 7726 8593 2112. The sum produced by the
algorithm is 65, found as follows:
- Double the even-positioned digits when counting from left to right
(1, 2, 9, 8, 2, 7, 1, 5). This results in 2, 4, 18, 16, 4, 14, 2, 10.
Four of these results are two-digit numbers; in those cases, add the
digits. Then, the eight numbers to be included in the sum are 2, 4, 9 (=1+8), 7 (=1+6), 4, 5 (=1+4), 2, 1 (=1+0).
- Add the results from Step 1 to the unaffected digits from the original number:
2 + 2 + 1 + 4 + 3 + 9 + 5 + 7 + 6 + 4 + 7 + 5 + 4 + 2 + 3 + 1= 65
- The sum is not a multiple of 10.
To be a valid account number, this sum must be evenly divisible
by 10. If the check digit were 7, the result would be congruent to 0
mod 10; but because the check digit is 2, the sum is not divisible
by 10. Therefore, this account number is not valid.
The Luhn algorithm is able to detect single data entry errors
and most transpositions. Students should proceed to the worksheet and
determine how this happens.
Prior to beginning the lesson, you may wish to review the solutions.
Questions for Students
1. A problem of the UPC system is that if two adjacent digits that were transposed have a difference of 5, the error will not be detected. Explain why this occurs.
[When the original digits are multiplied by 1 and 3 and the transposed digits are multiplied by 1 and 3, the difference of the two sums is 10. This is a problem because the sums of both UPC numbers will yield a remainder of zero when divided by 10.]
2. As we have seen on many television commercials, there are many banking institutions that offer credit cards. The first six digits that appear on a credit card are used for the issue identifier. How many possible issuers are there given each digit 0-9 could be used more than once?
[There would be 106 or 1,000,000 possible issue identifiers.]
3. This process is also used to detect most digit transpositions. For instance when entering a number 5832403 the data entry error is transposing the second and third digits: 5382403. There are two digits, when transposed, that will go undetected using the Luhn algorithm. What are they? Explain why this error cannot be detected.
[The digits that cannot be detected are 0 and 9. These are unique because the value of these two digits will always be a 0 and 9 regardless of their position in the account number. If the 9 is in the even-numbered position it will be doubled resulting in 18 with a sum of 1+8=9 and the 0 in the odd-numbered position would be unaffected. On the other hand, if 0 were in the even-numbered position, its value doubled would still be 0 and when added to the 9 the sum is still 9. Either way the sum of the two is nine.]
- How did the students demonstrate understanding of the materials presented?
- This lesson shows the importance mathematics plays in
something that millions of people use each day. Did the students gain
an appreciation for mathematics? Were they interested in learning about
how math is used in part of everyday life?
- What other lessons could be developed that would demonstrate the practically of math?
- What grouping approach did you choose for this lesson?
Partners, groups of 3 or 4? Was this approach effective? Why or why
not? What would you change for next time?
- Were concepts presented too abstractly? Too concretely? How would you change them?
- Did you find it necessary to make adjustments while teaching
the lesson? If so, what adjustments, and were these adjustments