Pascal's Triangle

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 Addressed

Grade 9

  • Estimation and Computation

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

  • Numeration

    • The student demonstrates conceptual understanding of real numbers.

Grade 10

  • Estimation and Computation

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

  • Numeration

    • The student demonstrates conceptual understanding of real numbers.

Fourth Grade

  • Operations and Algebraic Thinking

    • Generate and analyze patterns.

Fifth Grade

  • Operations and Algebraic Thinking

    • Analyze patterns and relationships.

Sixth Grade

  • The Number System

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

Algebra

  • Arithmetic with Polynomials and Rational Expressions

    • Use polynomial identities to solve problems

Technical Mathematics I

  • Number and Operations

    • Competency Goal 1: The learner will apply various strategies to solve problems.

Geometry

  • Geometry

    • Standard G-2: The student will demonstrate through the mathematical processes an understanding of the properties of basic geometric figures and the relationships between and among them.

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.

6th Grade

  • Number and Number Sense

    • 6.3a The student will find common multiples and factors, including least common multiple and greatest common factor
    • 6.3b The student will identify and describe prime and composite numbers; and identify and describe the characteristics of even and odd integers.

Textbooks Aligned

Grade Six

  • Prime Time

    • Investigation Two: The Product Game

Student Prerequisites

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

Teacher Preparation

Key Terms

combinatorics

The science that studies the numbers of different combinations, which are groupings of numbers. Combinatorics is often part of the study of probability and statistics

fractal

Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration

multiples

The product of multiplying a number by a whole number. For example, multiples of 5 are 10, 15, 20, or any number that can be evenly divided by 5

quotient

When performing division, the number of times one value can be multiplied to reach the other value represents the quotient. For example, when dividing 7 by 3, 3 can be multiplied twice, making 6, and the remainder is 1, so the quotient is 2

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

  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:

  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.

  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 Outline

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

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 Infinity, 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.