In this lesson, students will use clock arithmetic to find remainders. Then, they'll find patterns
in Pascal's triangle by coloring the remainders different colors.

This lesson is designed to follow the
Modular Arithmetic lesson. However, these lessons can be taught consecutively in a 2 hour block.

Objectives

Upon completion of this lesson, students will:

understand the relationship between remainders and clock arithmetic.

have practiced identifying and determining patterns in Pascal's Triangle.

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 6

Numeration

The student demonstrates conceptual understanding of fractions (proper or mixed numbers), decimals, percents (whole number), or integers.

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

Operations and Algebraic Thinking

Analyze patterns and relationships.

Fourth Grade

Operations and Algebraic Thinking

Use the four operations with whole numbers to solve problems.

Gain familiarity with factors and multiples.

Generate and analyze patterns.

Sixth Grade

The Number System

Compute fluently with multi-digit numbers and find common factors and multiples.

Third Grade

Operations and Algebraic Thinking

Represent and solve problems involving multiplication and division.

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.

6th Grade

Numbers and Operations

The student will demonstrate through the mathematical processes an understanding of the concepts of whole-number percentages, integers, and ratio and rate; the addition and subtraction of fractions; accurate, efficient, and generalizable methods of multiplying and dividing fractions and decimals; and the use of exponential notation to represent whole numbers.

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:

perform integer arithmetic.

understand how to find remainders using modular arithmetic.

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

pattern

Characteristic(s) observed in one item that may be repeated in similar or identical manners in other items

Lesson Outline

Focus and Review

Remind students what has been learned in previous lessons about remainders and modular arithmetic.

Ask students how to find the remainder of a given division problem, reminding them that they
can use modular arithmetic.

Ask the students to list the numbers that have a remainder of 1 when divided by 5.

Discuss patterns that they find as they do this.

Objectives

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

Today, we're going to look for remainders in a famous set of numbers.

As we do this, we're going to look for any patterns that we might find.

Teacher Input

Introduce Pascal's Triangle by playing a game. Explain the rules:

Divide the students into two or three groups.

Explain that there is a pattern in Pascal's Triangle, and you want them to complete it.

Explain that one (or two) students from each team come up at a time.

Each student will try to complete one line of the pattern in their team's triangle. Then the
teacher will check it.

If it is correct, the student will have a seat and the next teammate will try to complete
the next line.

If the line is incorrect, the teacher will erase it. Then the next student will try to
complete that same line.

Once the game is over, make sure everyone understands that each number is the sum of the two
numbers above it.

Guided Practice

Now that we're familiar with Pascal's Triangle, we're going to look for other patterns. One of
these patterns can be found by coloring remainders.

Now explain that you're going to use the
Clock Arithmetic applet to help you color the remainders.

Ask if anybody has any ideas of how you could use the applet to color the remainders.

Fill out the same triangle, using a divisor of 5. This time, use the clock arithmetic applet
to help.

On the Clock Arithmetic applet, set the clock size to 5 (or whatever your divisior is).

Set the start time at 0.

Put the number you're testing into the number of elapsed hours.

The number that the clock ends on will be the remainder.

Reset the clock and test the next number.

When dealing with larger numbers, be sure to set the clock to "Do Not Animate", so you
don't have to watch upwards of 50 rotations of the clock.

Now increase the depth of Pascal's Triangle.

Explain to the students that the Clock Arithmetic applet might be especially helpful when
working with really big numbers.

Independent Practice

Allow the students to work in pairs to fill out the Coloring Remainders in Pascal's Triangle
applet. If students finish early, have them fill the triangle out using a different divisor.

Closure

Lead the class in a discussion of patterns that they found. Ask the students why they think those
patterns occurred.

Alternate Outline

This lesson can be rearranged in the following ways if only one computer is available:

The teacher can instruct the students using a computer and projector and let them color the
remainders using a paper copy of pascal's triangle.

Students who need extra help with finding the remainders, can use the clock arithmetic applet
to color in Pascal's triangle.