Pascals Triangle Worksheet for 3rd - 5th Grade | Lesson Planet - Free Printable
Educational worksheet: Pascals Triangle Worksheet for 3rd - 5th Grade | Lesson Planet. Download and print for classroom or home learning activities.
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Step-by-step solution for: Pascals Triangle Worksheet for 3rd - 5th Grade | Lesson Planet
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Show Answer Key & Explanations
Step-by-step solution for: Pascals Triangle Worksheet for 3rd - 5th Grade | Lesson Planet
The image you uploaded is a worksheet about Pascal's Triangle, a triangular array of numbers where each number is the sum of the two directly above it. The task involves exploring patterns within Pascal's Triangle. Below, I will solve the problems step by step and explain the solution.
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- Solution: In Pascal's Triangle, the first occurrence of the number 5 is in the 6th row (counting the top row as row 0). It appears twice in that row, once on the left side and once on the right side.
- Row 6: \(1, 6, 15, 20, 15, 6, 1\)
- The first 5 appears in the 6th row.
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- Solution:
- The second 1 down on the right side is in the 2nd row (row index 1).
- Starting from this 1, we read the numbers in a diagonal going down and to the left:
- Row 2: \(1, 2, 1\) → The 1 is at position (1, 2).
- Row 3: \(1, 3, 3, 1\) → The next number in the diagonal is 3.
- Row 4: \(1, 4, 6, 4, 1\) → The next number in the diagonal is 6.
- Row 5: \(1, 5, 10, 10, 5, 1\) → The next number in the diagonal is 10.
- Row 6: \(1, 6, 15, 20, 15, 6, 1\) → The next number in the diagonal is 15.
- The pattern of numbers in this diagonal is: \(1, 3, 6, 10, 15, \ldots\), which are the triangular numbers.
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- Solution:
- The 3rd 1 from the top on the left side is in the 3rd row (row index 2).
- Starting from this 1, we read the numbers in a diagonal going down and to the right:
- Row 3: \(1, 3, 3, 1\) → The 1 is at position (2, 0).
- Row 4: \(1, 4, 6, 4, 1\) → The next number in the diagonal is 4.
- Row 5: \(1, 5, 10, 10, 5, 1\) → The next number in the diagonal is 10.
- Row 6: \(1, 6, 15, 20, 15, 6, 1\) → The next number in the diagonal is 20.
- The pattern of numbers in this diagonal is: \(1, 4, 10, 20, \ldots\), which are the tetrahedral numbers.
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- Solution:
- Let's choose the pair \(3\) and \(3\) from the 4th row (row index 3).
- The number just below them in the 5th row (row index 4) is \(6\).
- This demonstrates the property that each number in Pascal's Triangle is the sum of the two numbers directly above it.
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- Solution:
- To find the numbers in the 8th, 9th, and 10th rows, we use the property that each number is the sum of the two numbers directly above it.
- Eighth row (Row 7):
\[
1, 7, 21, 35, 35, 21, 7, 1
\]
- Ninth row (Row 8):
\[
1, 8, 28, 56, 70, 56, 28, 8, 1
\]
- Tenth row (Row 9):
\[
1, 9, 36, 84, 126, 126, 84, 36, 9, 1
\]
---
\[
\boxed{1, 7, 21, 35, 35, 21, 7, 1}
\]
\[
\boxed{1, 8, 28, 56, 70, 56, 28, 8, 1}
\]
\[
\boxed{1, 9, 36, 84, 126, 126, 84, 36, 9, 1}
\]
---
Problem 1: Find the first 5 in Pascal's Triangle
- Solution: In Pascal's Triangle, the first occurrence of the number 5 is in the 6th row (counting the top row as row 0). It appears twice in that row, once on the left side and once on the right side.
- Row 6: \(1, 6, 15, 20, 15, 6, 1\)
- The first 5 appears in the 6th row.
---
Problem 2: Find the second 1 down on the right side, and read the numbers in a diagonal going down and to the left
- Solution:
- The second 1 down on the right side is in the 2nd row (row index 1).
- Starting from this 1, we read the numbers in a diagonal going down and to the left:
- Row 2: \(1, 2, 1\) → The 1 is at position (1, 2).
- Row 3: \(1, 3, 3, 1\) → The next number in the diagonal is 3.
- Row 4: \(1, 4, 6, 4, 1\) → The next number in the diagonal is 6.
- Row 5: \(1, 5, 10, 10, 5, 1\) → The next number in the diagonal is 10.
- Row 6: \(1, 6, 15, 20, 15, 6, 1\) → The next number in the diagonal is 15.
- The pattern of numbers in this diagonal is: \(1, 3, 6, 10, 15, \ldots\), which are the triangular numbers.
---
Problem 3: Find the 3rd 1 from the top on the left side, and read the numbers in a diagonal going down and to the right
- Solution:
- The 3rd 1 from the top on the left side is in the 3rd row (row index 2).
- Starting from this 1, we read the numbers in a diagonal going down and to the right:
- Row 3: \(1, 3, 3, 1\) → The 1 is at position (2, 0).
- Row 4: \(1, 4, 6, 4, 1\) → The next number in the diagonal is 4.
- Row 5: \(1, 5, 10, 10, 5, 1\) → The next number in the diagonal is 10.
- Row 6: \(1, 6, 15, 20, 15, 6, 1\) → The next number in the diagonal is 20.
- The pattern of numbers in this diagonal is: \(1, 4, 10, 20, \ldots\), which are the tetrahedral numbers.
---
Problem 4: Choose a pair of side-by-side numbers in the triangle, then look at the number just below them
- Solution:
- Let's choose the pair \(3\) and \(3\) from the 4th row (row index 3).
- The number just below them in the 5th row (row index 4) is \(6\).
- This demonstrates the property that each number in Pascal's Triangle is the sum of the two numbers directly above it.
---
Problem 5: Write the numbers that would be in the eighth, ninth, and tenth rows of the triangle
- Solution:
- To find the numbers in the 8th, 9th, and 10th rows, we use the property that each number is the sum of the two numbers directly above it.
- Eighth row (Row 7):
\[
1, 7, 21, 35, 35, 21, 7, 1
\]
- Ninth row (Row 8):
\[
1, 8, 28, 56, 70, 56, 28, 8, 1
\]
- Tenth row (Row 9):
\[
1, 9, 36, 84, 126, 126, 84, 36, 9, 1
\]
---
Final Answer
\[
\boxed{1, 7, 21, 35, 35, 21, 7, 1}
\]
\[
\boxed{1, 8, 28, 56, 70, 56, 28, 8, 1}
\]
\[
\boxed{1, 9, 36, 84, 126, 126, 84, 36, 9, 1}
\]
Parent Tip: Review the logic above to help your child master the concept of pascal triangle worksheet with answers.
