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 10

Estimation and Computation

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

Numeration

The student demonstrates conceptual understanding of real numbers.

Grade 9

Estimation and Computation

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

Numeration

The student demonstrates conceptual understanding of real numbers.

Algebra

Arithmetic with Polynomials and Rational Expressions

Use polynomial identities to solve problems

Fifth Grade

Operations and Algebraic Thinking

Analyze patterns and relationships.

Fourth Grade

Operations and Algebraic Thinking

Generate and analyze patterns.

Sixth Grade

The Number System

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

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.

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:

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:

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.