Using Patterns to Solve Math Problems

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will learn how to use patterns to solve math problems by identifying patterns in real world data and extending those patterns to answer questions about different scenarios.


Suppose you decided to sign up for a half marathon (13.1 miles) race. After a long and grueling three months of training with a coach, race day is here! The gun goes off, and you embark on attempting a really impressive feat. Your coach records your elapsed time, in minutes, for the first five miles of the race, and the results are as follows:

Mile Elapsed Time
1 10
2 20
3 30
4 40
5 50

Do you notice any patterns in this chart? A pattern can be defined as a repeated occurrence, or as something that follows a specific rule. Hmm, well there are quite a few in there! For example, here are a few that you may have observed.

  1. Every time the miles go up by 1, the elapsed time goes up by 10.
  2. The elapsed time can be found by multiplying the mile number by 10.
  3. There is a difference of 1 between each mile in the chart, and there is a difference of 10 between each elapsed time in the chart.

That's just to name a few. We can observe patterns visually, numerically, or algebraically within this chart. Patterns, in general, show up all around us in the real world, and as it turns out, they're quite useful!

Using Patterns to Solve Math Problems

Let's get back to your race. After all, you're huffing and puffing away, and your coach is quite proud of you! Before the race, you made a goal of finishing the race in under 2.5 hours, or 150 minutes. Your coach is trying to figure out if you are on track to do that or if you need to speed up. I've got great news! Your coach can use the patterns in the chart to solve this problem.

There are a number of ways that your coach can use patterns to figure this out. He could extend the pattern out until the end of the race and see if your time is under 150 minutes at that point. He could also use the second pattern we named; that the elapsed time can be found by multiplying the mile number by 10 to set up the equation

T = 10m

where T is the elapsed time and m is the number of miles you've run.

Since he wants to know what your elapsed time would be at 13.1 miles, if you continue at the pace you are at, he can simply plug m = 13.1 into the equation and solve for T.


We see that if you continue at the pace you are at, then by extending the pattern, we find that at 13.1 miles, you will be a little over 130 minutes. To be more exact, the equation shows that you would finish in 131 minutes, or 2 hours and 11 minutes. Wow, speedy! You're going to crush your goal! Way to go!

This is a great example of using patterns to solve math problems. Just like we saw in this example, there are a number of ways to use patterns to solve problems such as extending the pattern or setting up an equation. Regardless of the exact method you choose, solving math problems using patterns involves these two steps:

  1. Identify patterns that your data seems to take on.
  2. Extend these patterns visually, algebraically, numerically, etc. to solve your problem.

Therefore, a very important part of using patterns to solve problems involves identifying and extending the patterns. Practice makes perfect, so let's consider another example.


The area of a square, given the length of the side of the square, is displayed in the chart.

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