Pascal's Triangle

The Chinese discovered this triangle in about 1100 AD but it wasn't until Blaise Pascal, a French mathematician, wrote his treatise in 1655 that the world knew about this triangles incredible importance. In China, they do not call it Pascal's Triangle, they call it Yang Hui's Triangle after the man who discovered it in China. But as usual, Pascal gets all the credit.

Pascal's Triangle is formed by starting with an apex of 1. Every number below in the triangle is the sum of the two numbers diagonally above it to the left and right. Each cell in the upper sides or legs of the triangle all contain ones.

So, the zeroth row of the triangle is 1, the first is 1 1, the second is 1 2 1, the third row is 1 3 3 1, the fourth row is      1 4 6 4 1, the fifth row is 1 5 10 10 5 1, and so on.

I gave the kids each an 11x17 blank of the triangle to 15 rows on one side and to 29 rows on the other side.

On the 15-row side (FOR ALL GROUPS EXCEPT FOR THURSDAY 4PM K CLASS), they are to add each two adjacent cells to fill in the cell below those two. For example, if two consecutive cells are 21 and 35, I had them write those two numbers down vertically, lining up the ones column (or units digits) and then add the ones column first followed by the tens column. Of course, they would get 56. The difficulty was that they were able to get to the first 5 or 6 rows by doing the math in their head. Then they resisted writing the numbers vertically. Every time they asked for help, I would ask them to write the numbers vertically and then walk them through the standard algorithm for addition.

Every time they wrote a number in the wrong place value, I showed them that their answer was the correct sum for a different number. For example, if they were adding 28 and 8 and put the 8 under the 2, they would get 108 because they were really adding 28 to 80. The next challenge was having them add the units digits first and place the units digit answer below the equals sign and carry the tens digit above the tens place value. Once they see this algorithm in action enough times they will own the concept. Even if they can try to solve these in their head, you should try to encourage them to write the numbers and add vertically.

I have attached an answer key for the first 15 rows. The frustrating thing is if they make a mistake, every number in the triangle below that number is also wrong. Get a good eraser.

On the 29-row side (FOR ALL GROUPS INCLUDING THURSDAY 4PM K CLASS), they are to add each two adjacent cells to fill in the cell below those two; except in this challenge they are to only record the units digit. For example, 3 + 3 = 6 (as you would for any Pascal's Triangle); 5 + 7 = 12 (but here you only record the units digit 2); 4 + 6 = 10 (so you record only the 0); 8 + 8 = 16 (so you record only the 6). This is great practice for quick addition using mental math.

Next week we will explore many of the famous patterns that were discovered by Pascal and a few that my  Mathletes have discovered over the years.