Pascal's triangle

Montessori Life, Spring 2003 by Barton, Patricia Shea

NEW IDEAS

"I am going to name my first child Trinomial Cube," one 4-year-old told his mother. He sometimes spent the entire morning working with the trinomial cube.

Another 4-year-old discovered the binomial cube within the trinomial as she was helping a younger child. I watched several 5- and 6-year-olds complete the trinomial cube with a blindfold, without hesitation.

The binomial and trinomial cubes were in almost constant use. This class was ready to continue on to other challenges of complex puzzles. Introduction to Pascal's Triangle provided another opportunity to connect young children to deep mathematical truths.

The Material

For the young child, Pascal's Triangle is a pattern related to logic. The materials include a black mat and 36 square tiles9 black and 27 white. The triangle is built from the top point down. The outline of the completed triangle is drawn on the mat.

After laying out the tile, I explain, "Always start with a white tile." This is the top point of Pascal's triangle. I then further explain, "We are going to build a triangle, Pascal's Triangle. There are two rules:

* If the tiles above are the same color, place a black tile below.

* If the tiles are different colors, place a white tile below."

Tile is added in rows from the left to right, from the top row down. Each new row begins one half of a tile's width to the left of the row above. Each row has one more tile than the row above. Tile placement is based on the colors of the two tiles in the row above that it will touch when the tile is in place.

Always start with the white tile. I point to the white tile and the black square on the mat next to it and say, "These are different, so I will place a white tile under them." Place a white tile under the previous tile. The first and second rows contain all white tiles. The next row contains the pattern white, black, white. You can continue indefinitely, but because of space and attention limitations, it works well to finish with Row 8-all white. This is the first iteration of Pascal's Triangle (Figure 1).

As with most new materials in the classroom, many children were interested. Children need to have a sense of order and be able to match, grasp, and sort. I showed the material to children of all ages. Many children returned later for clarification on the rules. Eventually I included with the work a guide illustrating the rules. It is still a challenge for many of the children.

What It Teaches

Through building the triangle, the child is introduced to a new way of looking at patterns. The activity requires him or her to create a pattern that is more than ABBA or ABAB. The form introduces the child to a rule-based pattern while still including some control of error. The child can recognize its balance and symmetry. Pascal's Triangle is an excellent entry point into mathematics at a variety of levels. Children are introduced to complex geometry and algebra visually by completing the repeating pattern. As they grow, they can create the numeric basis for the triangle through simple addition. They can recreate the pattern when they understand the concept of odd-and-even.

The Connection

Pascal's Triangle has a numeric foundation. In the preschool classroom, the children use a color representation. Follow along with the diagram in Table 1 (it will help to get apiece of paper to follow along).

Just as with the work, start by writing a 1. Now let's construct the next row. To the left of the 1 is an empty space, so consider it a zero. 0 1=1. Therefore, write a 1 to the left of the first 1, but in the second row. A similar process gets another 1 in the second row. Variation begins in the third row. Space plus one gives us the first numeral, but the middle numeral is 2. After that, increasingly varied patterns emerge. The fourth row is 1,3,3,1; the fifth is 1,4,6,3,1; and so on. Once you have created 8 rows, you can find the expression of Pascal's Triangle in the material. White tiles represent odd numbers; black tiles represent even numbers.

The numeric form of Pascal's Triangle is a mathematical wonder. It is filled with hidden relationships and connections to deeper mathematical concepts. Forexample, the sum of each row equals the powers of 2. The powers of 11 are found by combining placement in a row that represents a power of 10, so row 1=11(1) (see Table 2 for some of the other connections in the triangle) (Math Forum, 1998). The puzzles and connections provide late-elementary students the opportunity to discover mathematics in the same way preschool children are encouraged through sensorial materials.

Visual Information

Beyond the numbers and within the pattern, Pascal's Triangle is a related fractal, Sierpinski's Triangle. Fractals are complex mathematical relations found in nature. When a part of a fractal is enlarged or magnified, it produces a similar shape or pattern. Fractals are examples of mathematical beauty. They can be introduced visually at the preschool level.

Sierpinski's Triangle can be introduced in parallel to Pascal's Triangle. Children begin with a large equilateral triangle. The process of magnification is represented by placing greater and greater detail on the original triangle. Place four component equilateral triangles on top (Lanius, 2002). This is similar to the red equilateral triangle components in the triangular Constructive Triangle Box. Remove the middle equilateral triangle and replace it with another color. Repeat this with the remaining equilateral triangles (Figure 2). Fractals that are more complex could be introduced eventually.