1 - 2 class periods
- Ability to add like terms
- Experience in extending patterns
Ohio Standards Alignment
- Overhead transparency, activity sheet, and Pascal's Triangle template (all provided)
Simplifying expressions, Pascal's Triangle, generalizing patterns
Students complete a triangular array by inserting numbers into the top row and adding pairs of numbers for subsequent rows to achieve a final sum at the bottom. After discovering that the position of a number in the top row affects the final sum, they are challenged to find five different numbers that lead to a sum of 100 in the bottom row. They then analyze the diagram algebraically to determine a formula for the final result in terms of the starting numbers. Next, they compare the coefficients in their formula to the corresponding row of Pascal’s triangle and use that pattern to predict and complete a larger array with a specified end sum.
- To add polynomials, collect like terms, and simplify expressions.
- To discern and apply a pattern through Pascal’s triangle.
- To generalize and extend a pattern.
A triangular array (see Overhead) shows five circles in the top row, four in the next row, and so on, down to one circle in the last row. Hand out the Activity Sheet and ask students to fill in the rows in the first array in this way:
1. Write five different numbers in the top row.
2. For the next row, in each circle write the sum of the two numbers in the top row diagonally above the circle.
3. Complete the next three rows in the same way until you have the result in the bottom circle.
Can you write five different numbers in the top row so the result at the bottom is 100?
How does the arrangement of the numbers in the first row affect the sum at the bottom?
Analyzing the Pattern
To discover a formula for the result in the bottom circle, have students fill in the letters a, b, c, d, e into the diagram and write the polynomial sum in each circle in the second row, continuing until they complete the diagram.
Does your "formula" for the result at the bottom work for the numbers you used in the Critical Question above? Does it work for other numbers that you choose for the five starting numbers in row 1?
Pascal’s Triangle is an inverted triangular array of numbers that starts with 1 at the top. Each new row starts and ends with 1. Any numbers in between are the sum of the two numbers diagonally above them in the preceding row. Here are the first five rows:
1 2 1
1 3 3 1
1 4 6 4 1
What do you observe about the numbers in the fifth row and the formula you discovered above?
Generalizing the Pattern
Have students complete two more rows of Pascal’s Triangle, then ask the question:
How can you use the numbers in row 7 to complete a triangular array of single-digit whole numbers in row 1 so the result in the bottom circle is 253?
To answer this question, students could follow these guidelines:
a. Show Pascal’s Triangle for 7 rows.
b. Draw a triangular array with 7 rows of circles at the top and write a formula for the result in the bottom row.
c. Choose 7 different whole numbers for the circles in the top row and show how your completed diagram results in the number 253 at the bottom.
At the conclusion of this activity, discuss with students how algebra explains how numbers “behave” by using letters to represent generalized numbers and then following those general numbers through the various steps to determine how each step affects the end result. In this case, the pattern observed provides a formula for the final result, which is much easier to adjust trial numbers in than working through the whole diagram.
Pascal’s Triangle is an abstract generalization for the binomial behavior of the array, and one that is frequently encountered in mathematics. An extension of this activity is to look for the many patterns that can be discerned from Pascal's Triangle and how these patterns correspond to particular kinds of numbers (e.g., triangular numbers, pentagonal numbers).
Solutions to the questions can be found here.
See the following URLs for many intriguing patterns than can be found in Pascal’s Triangle:
Link to Pascal’s Petals: http://mathforum.org/workshops/usi/pascal/petals_pascal.html
Link to Twelve Days of Christmas and Pascal’s Triangle: http://dimacs.rutgers.edu/~judyann/LP/lessons/12.days.pascal.html
Link to Pascal's Triangle lessons: http://mathforum.org/workshops/usi/pascal/
Check this applet: http://www.ies.co.jp/math/java/misc/PascalTriangle/PascalTriangle.html
From the teaching files of Steven P. Meiring.