Abstract
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 3
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Numeration
- The student demonstrates conceptual understanding of whole numbers up to one thousand.
Grade 4
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Numeration
- The student demonstrates conceptual understanding of whole numbers to ten thousands.
Grade 5
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Numeration
- The student demonstrates conceptual understanding of whole numbers to millions.
Grade 9
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Estimation and Computation
- The student accurately solves problems (including real-world situations).
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Numeration
- The student demonstrates conceptual understanding of real numbers.
Grade 10
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Estimation and Computation
- The student accurately solves problems (including real-world situations).
-
Numeration
- The student demonstrates conceptual understanding of real numbers.
Grade 3
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Number Sense
- 2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division
Grade 4
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Number Sense
- 3.0 Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations
Third Grade
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Number and Operations in Base Ten
- Use place value understanding and properties of operations to perform multi-digit arithmetic.
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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.
Fourth Grade
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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.
Fifth Grade
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Number and Operations in Base Ten
- Understand the place value system.
Grades 3-5
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Numbers and Operations
- Compute fluently and make reasonable estimates
Grade 3
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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
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Number, Operation, and Quantitative Reasoning
- 4. The student recognizes and solves problems in multiplication and division situations.
Grade 4
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Number, Operation, and Quantitative Reasoning
- 4. The student multiplies and divides to solve meaningful problems involving whole numbers.
Grade 5
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Number, Operation, and Quantitative Reasoning
- 3. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
3rd Grade
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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
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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
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Arithmetic:
Students must be able to:
- complete basic whole number computations, including division with remainders.
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Technological:
Students must be able to:
- perform basic mouse manipulations such as point, click and drag.
- use a browser for experimenting with the activities.
Teacher Preparation
Students will need:
- access to a browser
- pencil and paper
- Copies of the Modular Arithmetic Exploration Questions
- Copies of the Working with Remainders Worksheet
- Working with Remainders Answer Sheet
Key Terms
division
The inverse operation of multiplication
modular arithmetic
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
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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.
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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.
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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.
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Guided Practice
Open a browser to the Clock Arithmetic applet.
- Demonstrate to the class how the applet works. Remember to explain that the clock can be used to show more than just elapsed time.
- As a class, brainstorm ways that you could use this applet to solve the problems from before.
- Have students use the applet to demonstrate these solutions to the rest of the class.
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Independent Practice
Have the students work on the Modular Arithmetic Exploration Questions.
- Have students discuss in groups how they solved the problems.
- Discuss as a class how to solve some of the more difficult problems.
Have the students complete the Working with Remainders Worksheet.
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Closure
Discuss as a class how using the Clock Arithmetic applet makes solving these problems easier.
Alternate Outline
This lesson can be rearranged in several ways if there is only one available computer:
- Have the students visualize modular arithmetic using the Clock Arithmetic applet but work through problems without a computer.
- Allow students who need extra help to use the Clock Arithmetic applet to help them solve the problems.
Suggested Follow-Up
This lesson can be followed by either of the following lessons:
- Finding Remainders in Pascal's Triangle instructs students on using modular arithmetic to look for patterns in Pascal's Triangle.
- Clock Arithmetic and Cryptography instructs students on how modular arithmetic and ciphers are linked, allowing students to create their own ciphers using modular arithmetic.