When first looking at Pascal's Triangle it may look like a simple triangle made up of adding numbers together and forming the shape of a triangle. But if you closely there are numerous patterns between the rows. One of my favorite patterns that I found in Pascal's Triangle is that of The Twelve Days of Christmas. This is a simple and very understandable way of explaining Pascal's Triangle.
There are four steps to this method. With the first being the number of new gifts given on the consecutive days. The first six steps are shown on the figure below. In Figure 1 you can see 1 turtle dove, 2 French hens, etc. I think this would be an excellent way to teach elementary aged children about Pascal's Triangle, especially with a popular Christmas song.
The second step in this process will combine the gifts given on each of the days.
For example on day 1 there is one partridge in a pear tree. By day two the true love has given 3 gifts, 2 turtle doves, 1 partridge. Now by day three the true love has given six gifts. The third step in this Christmas Triangle is that of the gifts given in total. For example day 1 the true love gives just the partridge in the pear tree. The next number in this pattern is a 4, which can be explained with 2 turtle doves + 1 partridge in a pear tree + 1 partridge in a pear tree given on the second day. This pattern can be seen in Figure 3 below. The next number here would be 10 which would be explained by 10 = 3 French hens + 2 turtle doves + 1 partridge in a pear tree + 2 turtle doves +...