Skip Counting with Pascal's Triangle

ND State Standard
2.NO.CC.1 Count forward from any given number within 1000.

2.NO.CC.2 Count backward from any given number within 1000.

2.NO.CC.4 Skip count forward and backward by 2s and 100s and recognize the patterns of skip counts

2.AR.OA.6 Identify a group of objects from 0 to 20 as even or odd by showing even numbers as a sum of two equal parts.

Items needed:

• Printed copy of the document embedded below or by clicking this direct link. 

  • You will need 3 copies of the 2nd page.

• Color crayon, marker, pencil to shade in certain numbers as outlined below. 

• Scissors

Pascal's Triangle is a number pattern where each number is the sum of the two numbers directly above it. It's a fascinating tool that can be used to explore various mathematical concepts. Let's see how it relates to our familiar friend, skip counting!

Build Pascal's Triangle

• Start with a 1 at the top.

• Every next row starts and ends with a 1.

• Each number inside the triangle is the sum of the two numbers directly above it.

• I've started these steps for you on each page, you'll need to finish through the 11th row provided.

Patterns

• Use the "Patterns" page.

• What patterns do you notice in the following diagonals?

  • Diagonal 1

  • Diagonal 2

• Horizontal Sums: Sum the numbers across each row.

  • Row 1 sum:

  • Row 2 sum:

  • Row 3 sum:

  • Row 4 sum:

  • Row 5 sum:

  • Row 6 sum:

  • Row 7 sum:

  • What pattern do you see within these sums? (this is well above a 2nd grade level but interesting!

Skip Counting by 2's

• Use the "Skip Counting by 2's Page"

• Color all the even numbers in the triangle, by skip counting by 2 to find each number.

• What do you notice about the positions of the even numbers? Are there any specific rows or diagonals where they appear consistently?

• Is there a relationship between the number of even numbers in a row and the row number? If so, what is the relationship?

Create the Sierpinski Triangle from Skip Counting by 2's

• If you have enough many more rows than we provide on our sheet and we went about, coloring the evens different colors results in pattern that look like a Sierpinski triangle. 

• We are going to create the Sierpinski Triangle by using 3 copies of your Skip Counting by 2's page. 

  • Print off 3 copies of the Skip Counting by 2's paper and color each one the same. 

  • Cut out Rows 1 through 8 as a single triangle.

    • Find out more about a Sierpinski Triangle from Math Is Fun.  By using rows 1 through 8 with the even numbers colored will provide the pattern in the 3rd step. 

  • Using 3 of these patterns will provide the 4th step, organize your 3 triangles to create the 4th step of the Sierpinski triangle and include a photo of this as part of your submission.

Skip Counting by 3's

• Use the "Skip Counting by 3's Page"

• Color all the numbers that are multiples of 3, by skip counting by 3 to find each number.

• Do you see any patterns in the positions of these numbers? Are there specific diagonals or triangular shapes formed by these multiples?

• Can you predict which rows will contain multiples of 3 without calculating them? How did you do this?

Skip Counting by 5's

• Use the "Skip Counting by 5's Page"

• Color all the numbers that are multiples of 5, by skip counting by 5 to find each number.

• What observations of particular patterns in the distribution of these numbers do you notice? Are there any specific conditions for a number to be a multiple of 5 in Pascal's Triangle?

Submission suggestions:

• You will want to include photo evidence of all the Pascal Triangle Patterns, Skip Counting and Sierpinksi Triangle.

• You will include your responses to the questions associated with the Patterns and Skip Counting by either a typed response or fully written response.