Coloring Remainders in Pascal's Triangle

What is Coloring Remainders 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
    

Related Resources

• Activity Coloring Multiples in Pascal's Triangle
• Activity Factorize
• Discussion about Pascal's Triangle
• Discussion about What are Remainders
• Worksheet Coloring Remainders Exploration Questions
• Worksheet Coloring Remainders Exploration Questions (doc)

How Do I Use This Activity?

This activity allows the user to visually identify number patterns in Pascal's triangle as they color remainders.

Controls and Output

• The Roll new value button randomly picks a number from 1 to the size of the triangle (including both numbers).
• The Auto-color! button automatically colors all the numbers in the triangle based on the current divisor number.
• The text field shows the value of the current divisor. This text field also allows the user to enter his or her own number to be set as the new divisor. Either press the return key or click the Set button after inputting the divisor.
• The current... message board provides the current divisor in case you forget.
• The Click... 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.
• 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 formed in Pascal's triangle when coloring remainders. This activity would work well in groups of two or three for about forty-five minutes if you use the exploration questions and twenty to twenty-five minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

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

Be Prepared to

• explain that they have to color all the numbers with one remainder before they color the numbers with a different remainder.
• explain how coloring remainders produces patterns similar to the ones noticed when coloring multiples in Pascal's triangle.

Related Resources

• Activity Coloring Multiples in Pascal's Triangle
• Activity Factorize
• Discussion about Pascal's Triangle
• Discussion about What are Remainders
• Lesson on Finding Remainders in Pascal's Triangle
• Lesson on Introduction to Fractals: Geometric Fractals
• Lesson on Pascal's Triangle
• Lesson on Patterns in Pascal's Triangle
• Worksheet Coloring Remainders Exploration Questions
• Worksheet Coloring Remainders Exploration Questions (doc)