Illuminations: Arithme-Tic-Toc

Arithme-Tic-Toc
Students will be introduced to modular arithmetic by first examining a five-hour analog clock and its mathematical properties. Then students will investigate patterns and relationships that exist in 12-hour addition and multiplication clock tables.

Learning Objectives
 
Students will:
  • Determine which properties of real numbers are also true for modular arithmetic.
  • Conjecture about patterns and other relationships that exist in the modulus tables.
  • Apply the modular arithmetic concept to everyday occurrences.

Materials
 
Arithme-Tic-Toc Activity Sheet
Analog Clock
Modulus Tables

Instructional Plan
Because of its familiarity to students, one simple way to introduce modular arithmetic is with a 12-hour analog clock.

Begin this lesson by writing 10 + 4 = 2 on the chalkboard. Most students will object to this untrue statement, but then ask, "Can you think of a situation in which this statement might be true?" There are a number of possible answers that students could suggest:

  • 10 a.m. + 4 hours = 2 p.m.
  • 10th grade + 4 years = 2nd year in college
  • 10 gross + 4 gross = 2 gross more than a great gross
  • 10 inches + 4 inches = 2 inches more than a foot
  • 10 points + 4 points = 2 points more than a pica (in printing)

Students might also suggest answers for which the units change on either side, such as 10 days + 4 days = 2 weeks. Because the question is open-ended, these answers are acceptable; but to lead into the day’s lesson on modular arithmetic, you will want to spend more time on examples like those in the list above.

Explain to students that many of the examples above deal with modular arithmetic, which is the mathematics of remainders.

To get students familiar with modular arithmetic, ask, "What time will it be six hours from 5 p.m.?" Students will quickly give the correct answer, 11 p.m. Then ask, "How about 14 hours from 5 p.m.? How would you determine the time?" Students will suggest that 7 hours will take the time to midnight, and the remaining 7 hours will take the time to 7 a.m. Challenge students to generalize this last question. Have them determine an algorithm that can be used to find the end time if 12 a.m. or 12 p.m. is passed. [Add the current time to the number of additional hours, divide by 12, and the remainder is the end time.]

For demonstration purposes, you may want to show an analog clock during this portion of the lesson. It may help to remove the minute and hour hands, so students focus on the mathematics of the lesson and not the time.

Using this discussion as a springboard, you may want to have students work in pairs to create a mod‑12 addition table. Alternatively, such a table is provided as the last sheet of the Arithme-Tic-Toc activity sheet.

Arithme-Tic-Toc Activity Sheet Arithme-Tic-Toc Activity Sheet

Students can also create mod‑12 multiplication tables. Alternatively, the Arithme‑Tic‑Toc Spreadsheet can be used by students to discover how multiplication works with modular arithmetic. With this spreadsheet, students can adjust the modulus, see the resulting table, and make some discoveries about the results.

Arithme-Tic-Toc Spreadsheet Arithme-Tic-Toc Spreadsheet (Excel)

The spreadsheet can be used to generate class discussion on pattern relationships that exist within each table. After allowing sufficient time for group discussion, solicit responses from students and list all observations on the board. As a class, discuss the observations, and allow students to discover any incorrect observations that were made. In addition, encourage students to give reasons for some of the things they noticed.

Once students understand basic computations with modular arithmetic, proceed to the handout where students will investigate relationships and discover interesting patterns. Prior to working on the handout, however, you may want to point out several properties regarding modular arithmetic and its notation:

  • Modular arithmetic uses integers only.
  • The notation a = b mod m implies that a and b have the same remainder when divided by m.
  • If a = b mod m, then a – b is a multiple of m.

Other properties may be reviewed and discussed, too. To make the discussion more interesting, rather than stating these properties, ask students questions that highlight these properties. For instance, you might show students the equations "31 = 3 mod 7" and "25 = 1 mod 8," and ask them to take a guess at what the notation means.


On the Arithme-Tic-Toc activity sheet, the last page contains mod‑12 addition and multiplication tables. Although students will not need them to complete the mod‑5 tables on the first page, students can use these tables to complete Questions 1‑10. (If you had students generate their own mod‑12 tables earlier in the lesson, students can use their own table to answer these questions.) Questions 1‑3 allow students to explore the addition table, and questions 4‑10 allow them to explore the multiplication table. Students will likely find the multiplication table more interesting, because there are more patterns to be discovered.

Answers to all of the questions on the activity sheet are provided in the Arithme-Tic-Toc Answer Key.

Questions for Students
 
What is the algorithm for addition in modular arithmetic?
[Add two numbers together, divide by the modulus, and the remainder is the answer.]

Take a look at the mod‑12 multiplication table. Notice that several rows and columns contain each of the values from 1 to 11 exactly once. Explain why this happens. (Hint: Think about factors.)

[This occurs in Rows 1, 5, 7, and 11. It is due to the fact that these numbers are relatively prime to 12.]

Assessment Options
 
  1. Assign student groups to create the multiplication and addition tables for moduli other than 5 and 12. Students can investigate their assigned tables for mathematical ideas and present their findings to the class.

  2. In whole‑class discussions, ask students to describe how they discovered their patterns. Encourage and validate a variety of appropriate responses.

  3. Using the activity sheet as a guide, ask students to write an entry in their journals that includes the following:
    • A summary of what they found, as well as why they think their findings are accurate.
    • Any tables, charts, or other tools they used to organize their information.
    • An explanation of any patterns they found.

Extensions
 
  1. As a journal writing activity and exploration, students can construct mod n tables for the operations of subtraction and division. Students should examine these tables to determine if any of the discoveries in the addition and multiplication tables apply, plus any other math ideas students can discover. Challenge students to find the equivalent clock values for negative integers. Specific examples along with a written explanation should be included.
    [Division provides some difficulties not found with the other operations. There is not always a unique answer, there may be multiple answers, or there may be no answer at all. To see this, remember that division and multiplication are inverse operations. Therefore, the answer to a division problem can be found by looking at the associated multiplication problem. For instance, using the mod‑12 multiplication table, determine the answer to the problem 9 ÷ 3 = w. We are therefore trying to find a value for w such that w × 3 = 9. From the chart, notice that 7 × 3 = 9, 11 × 3 = 9, and 3 × 3 = 9. Therefore, there is not a unique solution, and thus division is not uniquely defined. The numbers that are uniquely defined are those that are relatively prime to 12, namely 1, 5, 7 and 11.]

  2. Have students with electronic keyboards bring them to class and compose modular music. The basic structure for this is to base the melody on prime numbers and transfer that to the keyboard. Determine how many prime numbers you wish to use in your melody and which prime you will begin with. Then determine a modulus, n, and calculate the corresponding modular value for each prime number. For example, beginning at 2 and continuing with all primes through 29 mod 6 would give the pattern 2, 3, 5, 1, 5, 1, 5, 1, 5, 5. This pattern of numbers would become the keys that would be played.

  3. Prove that the mod‑12 multiplication table is associative.

Teacher Reflection
 
  • Did you remove the hour and minute hands when introducing the modular arithmetic clock? Was this an effective way to introduce this topic? How else could you introduce this topic?
  • Did the mod‑5 addition and multiplication tables provide a solid foundation for students to thoroughly investigate the mod‑12 tables? If not, what tables could be used?
  • This lesson introduces students to an unfamiliar topic. Did you find it too abstract for them? What could be done to address the different learning styles of students?
  • Were the questions challenging but not too difficult for students? Did you find that you had to provide much help, or were students able to work independently? If needed, what adjustments could be made to promote more student independence in completing the activity sheet?

NCTM Standards and Expectations
 
Algebra 9-12
  1. Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.
  2. Use symbolic algebra to represent and explain mathematical relationships.
Number & Operations 9-12
  1. Use number-theory arguments to justify relationships involving whole numbers.
  2. Judge the reasonableness of numerical computations and their results.
This lesson prepared by Doug Schmid.
  
2 periods   

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