# Identifying & Extending Different Types of Patterns

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will look at different ways patterns present themselves in the world around us. We will take a look at how to identify patterns and how to extend patterns and use this in some real-world examples.

## Identifying Patterns

Suppose me, you, Mary, and John all head out for a walk, and we notice that the path we are walking on seems to be marked for a running race that will be held tomorrow. As we are walking along, we see that after 0.25 miles, there is a line spray painted on the road. After 0.5 miles, there is a square. After 0.75 miles, there is a circle, and when we reach a mile, there is a star spray painted on the road. We continue walking for three miles and notice the same markings at the same intervals through each mile.

By observing these shapes on the side of the road and realizing that they repeat themselves every mile, we are actually identifying a pattern. In mathematics, identifying a pattern involves observing a phenomenon that occurs over and over again.

## Extending Patterns

Alright, we've gone three miles. I'd say that's a pretty decent distance! Let's agree to stop here. However, curiosity strikes! We've identified the pattern of the shapes on the side of the road, but we aren't going to continue our walk to see the pattern through to the end of the race. You see, Mary's favorite number is 4.25, and John's favorite number is 5.5, so they would like to know what shape would show up at those distances. Any ideas on how we could figure this out without having to continue our walk to those points?

If you are thinking we can use our pattern to figure this out, you are absolutely correct! If we continue the pattern we identified out to 5.5 miles along the course, then we will be able to see what shape is at 4.25 miles and at 5.5 miles!

We see that at 4.25 miles, there will be a line, and at 5.5 miles, there will be a square. This process of continuing the identified pattern is called extending the pattern. In general, extending a pattern involves first identifying the pattern and then continuing the pattern.

## More Examples

By identifying and extending a pattern, we were able to satisfy our curiosity. This is great, and what's even better is that this process of identifying and extending patterns shows up all the time in the world around us, so we can use what we've learned quite often! Let's consider some of these instances.

Suppose you are looking to invest in the stock market. Before doing so, you observe the trends that a particular investment option seems to take. You find that each month, the stock value seems to follow the pattern shown in the graph.

What patterns do you identify? It looks as though the stock value falls to a minimum at the beginning of each month, then spikes mid-month, and then evens out towards the end of the month. Obviously, it's best to buy when the value is low and sell when the value is high. Based on this, at what part of next month should you buy stock?

If we extend the pattern one more month, we observe that the stock is at its lowest at the beginning of the month.

Therefore, it would be in your best interest to buy at the beginning of the month. Pretty neat, huh?

Patterns present themselves in many different ways. Our marathon example is a pattern observed in pictures. The stock market example is one identified in graphs. Let's consider an example where a pattern presents itself numerically and algebraically.

Suppose Chuck is saving up to buy an engagement ring for his girlfriend. He is putting away \$500 each month, and the ring he would like to get costs \$3800. For the first 5 months, he makes a chart of how much money he has saved.

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