Always Check Your Work Exploration

First, add the numbers in each row of the triangle. Write down the sum of the first 8 rows.  We’ll call this pattern 1.

Do you notice anything interesting about the sums of the elements (numbers) in each row? Is there any pattern you can identify?

Alright, now another amazing pattern involves something we almost never do in math: pushing numbers together to make a new number.  For example, we almost never take the number 6 and the number 8 and just push them together to make 68, right?  Well, let’s try exactly that.  Write the numbers in each row as if they were the digits in a single number.  (Hint: 1, 11, 121…)  You can stop after 5 rows for now. 

Do you notice anything interesting about these numbers?  If not, try these problems in your calculator and compare it to these numbers you just wrote down. 

\( {{11}^{0}}\)                        \( {{11}^{1}}\)                            \( {{11}^{2}}\)                          \( {{11}^{3}}\)

Alright, let’s call this pattern 2.

Now we’ve explored a couple of the patterns that are relatively obvious in Pascal’s Triangle.  There are also tons of patterns that aren’t quite so obvious.  One such pattern is the third diagonal in the triangle (the first diagonal is just 1’s, the second diagonal is the natural numbers).  List the first 8 numbers in this pattern. 

Can you figure out what is special about this pattern? Now this is a tough pattern to identify but give it a shot. (Hint: Don’t be discouraged, “tri” your best)

This pattern is called the triangular numbers.  Why are they called triangular number? Because they make equilateral (equal length sides) triangles.  The triangular numbers are the number of pins you would need to set up perfect arrangements of bowling pins. Below, you can see how the 1 and 3 can be made into equilateral triangles.  Can you do the same with the rest of the triangular numbers up to 45?  Draw it out in your notebook and submit a scan or picture.

In Pascal’s Triangle below, the triangular numbers stop at 21.  One way to find more is to continue generating more rows of Pascal’s Triangle.  Fortunately, there is a pattern with the triangular numbers.  See if you can discover the pattern and determine the next 3 triangular numbers.

Let’s call the triangular numbers pattern 3.