Finding Remainders in Pascal's Triangle

Abstract

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.

Student Prerequisites

• Arithmetic: Students 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.

Teacher Preparation

Students will need:

• Access to a browser
• Pencil and paper
• Empty Pascal's Triangle

Lesson Outline

1. 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.

2. 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.

3. 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.

4. 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.

   Explain how the Coloring Remainders in Pascal's Triangle applet works. Begin with a divisor of 5. Fill out the triangle as a class.

   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.

5. 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.

6. 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.

Suggested Follow-Up

Students can further explore Pascal's Triangle in the Patterns in Pascal's Triangle lesson.