In this lesson, students will learn about modular arithmetic and how to apply it in real world
situations.

Objectives

Upon completion of this lesson, students will:

understand how to perform modular arithmetic.

understand the notation # mod #.

be able to apply modular arithmetic in real world contexts.

Standards Addressed:

Grade 10

Estimation and Computation

The student accurately solves problems (including real-world situations).

Numeration

The student demonstrates conceptual understanding of real numbers.

Grade 3

Numeration

The student demonstrates conceptual understanding of whole numbers up to one thousand.

Grade 4

Numeration

The student demonstrates conceptual understanding of whole numbers to ten thousands.

Grade 5

Numeration

The student demonstrates conceptual understanding of whole numbers to millions.

Grade 9

Estimation and Computation

The student accurately solves problems (including real-world situations).

Numeration

The student demonstrates conceptual understanding of real numbers.

Grade 3

Number Sense

2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division

Grade 4

Number Sense

3.0 Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations

Fifth Grade

Number and Operations in Base Ten

Understand the place value system.

Fourth Grade

Number and Operations in Base Ten

Generalize place value understanding for multi-digit whole numbers.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

Third Grade

Number and Operations in Base Ten

Use place value understanding and properties of operations to perform multi-digit arithmetic.

Operations and Algebraic Thinking

Represent and solve problems involving multiplication and division.

Understand properties of multiplication and the relationship between multiplication and division.

Multiply and divide within 100.

Grades 3-5

Numbers and Operations

Compute fluently and make reasonable estimates

Grade 3

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 1: The learner will model, identify, and compute with whole numbers through 9,999.

COMPETENCY GOAL 5: The learner will recognize, determine, and represent patterns and simple mathematical relationships.

Grade 4

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 1: The learner will read, write, model, and compute with non-negative rational numbers.

Grade 5

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 1: The learner will understand and compute with non-negative rational numbers.

COMPETENCY GOAL 5: The learner will demonstrate an understanding of patterns, relationships, and elementary algebraic representation.

Grade 3

Number, Operation, and Quantitative Reasoning

4. The student recognizes and solves problems in
multiplication and division situations.

Grade 4

Number, Operation, and Quantitative Reasoning

4. The student multiplies and divides to solve
meaningful problems involving whole numbers.

Grade 5

Number, Operation, and Quantitative Reasoning

3. The student adds, subtracts, multiplies, and
divides to solve meaningful problems.

3rd Grade

Patterns, Functions, and Algebra

3.25a The student will investigate and create patterns involving numbers, operations (addition and multiplication), and relations that model the identity and commutative properties for addition and multiplication.

5th Grade

Computation and Estimation

5.5 The student, given a dividend of four digits or fewer and a divisor of two digits or fewer, will find the quotient and remainder.

Student Prerequisites

Arithmetic: Student must be able to:

complete basic whole number computations, including division with remainders.

Technological: Students must be able to:

perform basic mouse manipulations such as point, click and drag.

use a browser for experimenting with the activities.

A method for finding remainders where all the possible numbers (the numbers less than the divisor) are put in a circle, and then by counting around the circle the number of times of the number being divided, the remainder will be the final number landed on

remainders

After dividing one number by another, if any amount is left that does not divide evenly, that amount is called the remainder. For example, when 8 is divided by 3, three goes in to eight twice (making 6), and the remainder is 2. When dividing 9 by 3, there is no remainder, because 3 goes in to 9 exactly 3 times, with nothing left over

Lesson Outline

Focus and Review

Remind students what has been learned in previous lessons that will be pertinent to this lesson
and/or have them begin to think about the words and ideas of this lesson:

Ask the students if they remember how to divide in situations such as 15/4.

Have students explain to one another how to divide with remainders.

Objectives

Let the students know what it is they will be doing and learning today. Say something like this:

Today, class, we will be talking about modular arithmetic and how to use it to solve real
world problems.

We are going to use the computers to learn about modular arithmetic, but please do not turn
your computers on or go to this page until I ask you to. I want to show you a little about
this activity first.

Teacher Input

Give the student this word problem: If I have 14 cookies, and I want to divide them evenly among 5
students, how many cookies would each person get?

Ask the students how to solve this problem, not what the answer is.

Write the names of 5 students on the board like a clock-face. Draw a cookie next to each name
as you "deal" out the cookies. Ask the students what the remainder is.

Explain to the students that 14 mod 5 is 4. Ask them what they think "mod" means. Then ask the
students what 25 mod 3 is.

Focus on the process of how they solved the question, not the answer.

Make sure everyone understands the process, perhaps letting students pair-share with each
other to solidify their understanding.

Clock Arithmetic and Cryptography instructs students on how modular arithmetic and ciphers are linked, allowing students to
create their own ciphers using modular arithmetic.