“Arithmetic begins with learning to count by ones, after that, it is a never-ending search for shortcuts to avoid one-by-one counting.”

Robert Wirtz
Drill and Practice at the Primary School Level

There are two kinds of patterns: repeating and growing.  Repeating patterns have a sequence that goes and then repeats itself over and over again.  Growing patterns have a starting point and then grow by a set interval infinitely. 

The power of patterns is that they allow us to predict what will come next and they allow us to solve problems that would be very tedious to solve otherwise. 

The story goes that a young boy walked into his class and read the assignment: Add up all the numbers from 1 to 100.  He quickly calculated in his head and said, “5050.” 

“That’s amazing!” his teacher exclaimed.  “How did you add them so quickly?”

“I didn’t add them,” the boy responded, “I saw the pattern.”

 The story is of course fictionalized, but it is based on the accounts of a true event involving Carl Gauss.  Apparently, he noticed that the lowest and highest numbers equaled 101 (1 + 100=101) and that this pattern repeated itself 50 times (2+99=101), (3+98=101…(50+51=101).  He then only needed to multiply 101 times 50 to solve the problem.  

Students need to learn to look for these patterns.  They help them to make connections, predict what will come next, and solve problems.  Math has been defined as the study of patterns.  We study patterns because they are everywhere; we just need to learn to notice them.