Interactivate: 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:

Textbooks Aligned:

Student Prerequisites

Arithmetic: Student 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

Access to a browser
Pencil and calculator
Copies of supplemental materials for the activities:

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

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:

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.

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.

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?"

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.

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:

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 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.

Interactivate

Pascal's Triangle