Abstract

The following discussions and activities are designed to lead the students to explore the number patterns and fractal properties of Pascal's Triangle. Basic arithmetic operations of multiplication and long division are practiced in a novel way.

Objectives

Upon completion of this lesson, students will:

• have learned about Pascal's triangle, including how to build it and a few of its uses
• have practiced their integer multiplication and division skills

Standards

The activities and discussions in this lesson address the following NCTM Standards:

Number and Operations

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

• use factors, multiples, prime factorization, and relatively prime numbers to solve problems

Algebra

Understand patterns, relations, and functions

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules

Represent and analyze mathematical situations and structures using algebraic symbols

• use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;

Links to other NCTM standards.

Student Prerequisites

• Arithmetic: Students must be able to:
  • understand and manipulate integers
  • perform simple multiplication and division of integers
• Algebraic: Students must be able to:
  • work with simple algebraic expressions (including integer powers)
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• Access to a browser
• Pencil and calculator
• Copies of supplemental materials for the activities:
  • coloring multiples worksheet
  • coloring remainders worksheet
  • Pascal's Triangle handout

Key Terms

• combinatorics
• fractal
• multiples
• quotient
• remainder

Lesson Outline

1. 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:

   • Geometric Fractals

   • Chaos Game

2. 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 number patterns and fractals while looking at Pascal's triangle.

3. Teacher Input

   You may choose to lead the students in short discussions on multiples and integer multiplication, remainders and Euclidean division, and / or Pascal's triangle.

   Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use our computer applets.

   • Open your browser to Coloring Multiples in order to demonstrate this activity to the students.

   • If you choose to, you may pass out the exploration questions for the Coloring Multiples activity.

4. Guided Practice

   Try an example coloring using a given number (say 2), letting the students direct your moves. Or, you may simply ask, "Can anyone describe the steps you will take for this assignment?"

   • If your class seems to understand the process for doing this assignment, start with another number (say 5) and simply ask, "Can anyone tell me what you will do now?"

5. Independent Practice
   • Allow the students to work on their own and to complete the worksheet, should you choose to provide one. Monitor the room for questions and to be sure that the students are on the correct web site.

   • Have the students try the computer version of the Coloring Remainders activity to investigate the patterns of the remainders in Pascal's triangle. The exploration questions could be handed out for students to work on independently.

6. Closure

   You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outlines

This lesson can be modified if there is only one available computer:

• Walk through the three discussions -- multiples and integer multiplication, remainders and Euclidean division, and / or Pascal's triangle.

• Have the students try coloring by hand, using copies of the paper version of Pascal's triangle.

• Use the computer activities -- Coloring Multiples and Coloring Remainders -- in demo mode to have the students check their answers. For example, students could be called to the front to check their colorings by bringing their sheet to the demo computer and entering their coloring.

Suggested Follow-Up

After these discussions and activities, the students will have seen more places in which fractal patterns similar to Sierpinski Triangles, introduced in the Geometric Fractals and Fractals and the Chaos Game lessons, appear. The next lesson, Irregular Fractals, generalizes fractals, as seen in the Self-Similarity and Recursion, Geometric Fractals, Fractals and the Chaos Game lessons, showing how they can be used to create pictures that look like natural phenomena.