Coloring Multiples in Pascal's Triangle

What is Coloring Multiples in Pascal's Triangle?

This activity allows the user to investigate number patterns in Pascal's Triangle created by placement of multiples.

Pascal's Triangle is a triangle of numbers, each new number being the sum of the two above it. Here are a few rows:

  • In combinatorics and counting, we can use these numbers whenever we need to know the number of ways we can choose Y things from a group of X things.
  • In algebra, these numbers are used to determine the coefficients on binomial raised to a particular power. The power is associated with the row in the triangle. The first row of the triangle is considered the zeroth row. For instance, 
         (a + b)4
    multiplied out in expanded form yields 
         a4 + 4a3b + 6a2b2 + 4ab3 + b4
    Notice how the 4th line of the triangle (counting the first row as 0) is 
         1  4  6  4  1
    And notice that the coefficients in the expanded form of the binomial are also 
         1  4  6  4  1

How Do I Use This Activity?

This activity allows the user to visually identify number patterns in Pascal's triangle as they color multiples of a given number.

Controls and Output

  • The Roll new value button randomly picks a number between (and including) 2 and 10. You need to pick which numbers in the triangle below are multiples of the specified number.
  • The Auto-color! button automatically colors all the numbers in the triangle below that are multiples of the specified number.
  • The text field shows what value you should be referring to when you are picking which numbers in the triangle below are multiples of the specified number. This text field also allows you to enter your own number. Set your own value by entering a value and then press the return key or click the Set button.
  • The message board provides instructions as well as comments on your progress.
  • The increase depth and decrease depth buttons allow you to increase and decrease the number of rows displayed in the triangle.
  • This activity will automatically record how successful you are at answering the questions. To view the score, press the Show Score button at the bottom of the activity and a pop-up window will appear with the scoreboard. To close this pop-up window press the Close button or click back on the main window.
  • To pause the scoring, press the Active button at the bottom of the screen and it will change to a Paused button. To resume scoring, press the Paused button.
  • To reset the scoreboard, open the scoreboard using the Show Score button and then press the Reset button.

Description

This activity allows the user to identify number patterns in Pascal's triangle. This activity would work well in groups of two or three for about forty minutes if you use the exploration questions and fifteen to twenty minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

  • practice students' multiplication skills
  • practice students' pattern recognition skills
  • introduce Pascal's triangle
  • motivate the ideas of fractals

Standards Addressed

Grade 3

  • Numeration

    • The student demonstrates conceptual understanding of whole numbers up to one thousand.

Grade 4

  • Numeration

    • The student demonstrates conceptual understanding of whole numbers to ten thousands.

Grade 5

  • Numeration

    • The student demonstrates conceptual understanding of whole numbers to millions.

Grade 6

  • Estimation and Computation

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

  • Numeration

    • The student demonstrates conceptual understanding of fractions (proper or mixed numbers), decimals, percents (whole number), or integers.

Grade 7

  • Estimation and Computation

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

Grade 8

  • Estimation and Computation

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

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.

Third Grade

  • Operations and Algebraic Thinking

    • Represent and solve problems involving multiplication and division.
    • Understand properties of multiplication and the relationship between multiplication and division.
    • Multiply and divide within 100.
    • Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Fourth Grade

  • Operations and Algebraic Thinking

    • Use the four operations with whole numbers to solve problems.
    • Gain familiarity with factors and multiples.
    • 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

Grades 3-5

  • Algebra

    • Understand patterns, relations, and functions
    • Use mathematical models to represent and understand quantitative relationships

  • Numbers and Operations

    • Compute fluently and make reasonable estimates

Grades 6-8

  • Numbers and Operations

    • Understand meanings of operations and how they relate to one another

Technical Mathematics I

  • Number and Operations

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

6th Grade

  • Numbers and Operations

    • The student will demonstrate through the mathematical processes an understanding of the concepts of whole-number percentages, integers, and ratio and rate; the addition and subtraction of fractions; accurate, efficient, and generalizable methods of multiplying and dividing fractions and decimals; and the use of exponential notation to represent whole numbers.

5th Grade

  • Algebra

    • The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

  • Number and Operation

    • The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

3rd Grade

  • Numbers and Operations

    • The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

4th Grade

  • Algebra

    • The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

  • Number and Operation

    • The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

Be Prepared to

  • explain how divisibility by a number is the same as being a multiple since the applet's name is coloring multiples but the directions say color the numbers divisible by the number rolled.
  • discuss Sierpinski's triangle.