# What is a Pattern in Math? - Definition & Rules

## What Is a Pattern?

'The itsy-bitsy spider went up the water spout. Down came the rain and washed the spider out. Out came the sun and dried up all the rain, and the itsy-bitsy spider went up the spout again.' The itsy-bitsy spider song is an example of a pattern. A **pattern** is a series or sequence that repeats. The itsy-bitsy spider climbed the water spout, and then did the same thing again after the weather cleared up.

You can observe patterns - things like colors, shapes, actions, or other sequences that repeat - everywhere. Think about words or melodies in songs, lines and curves on buildings, or even in the grocery store where boxes and jars of various items are lined up.

But, one of the most common places to find patterns is in math. **Math patterns** are sequences that repeat according to a rule or rules. In math, a **rule** is a set way to calculate or solve a problem.

## Number Patterns

One common type of math pattern is a number pattern. **Number patterns** are a sequence of numbers that are ordered based upon a rule. There are many ways to figure out the rule, such as:

- Use a number line to see the distance between the numbers or what they have in common
- Look at the last one or two digits or the first digit to see if they repeat in a special manner
- Look at the numbers and see if there is a pattern, like taking each number and multiplying by 3 for instance
- Think about common number patterns, like counting by 2s, 5s, or 10s, and/or
- Find the difference between the numbers

It's important to remember that a number pattern can have more than one solution and a combination of rules. If this is the case, try to think of the simplest rule possible, like adding 1 or multiplying by 2 with a difference of 3.

## Shape Patterns

Another common type of math pattern is a shape pattern. **Shape patterns** are exactly what they sound like - a sequence of shapes that are arranged based upon a rule. Instead of being identified by the types of shapes in the pattern, shape patterns are identified by using letters like A, B, or C. Each shape gets its own letter, but when the shape repeats, you must use the same letter again for that shape.

For example, the shape pattern in the picture starts with a triangle, so you will give each triangle the letter A. Next, there is a square which will get the letter B. Then, there's a circle, so it will get the letter C. Afterwards, we see another square, so you will give it the letter B, and the pattern repeats again starting with the triangle. This would be called an ABCB pattern, based on how the shapes repeat.

## Lesson Summary

Let's review by going back over all our terms for understanding patterns in math. A **pattern** is a series or sequence that repeats. **Math patterns** are sequences that repeat based on a rule, and a **rule** is a set way to calculate or solve a problem. There are two main types of math patterns: **number patterns**, or sequences of numbers arranged according to a rule or rules, and **shape patterns**, which are labeled by using letters and the way that they repeat.

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## Applications of Patterns in the Real World:

### Questions:

- At a bank, a customer deposits a specific amount in an account each week. The bank's computer system recently crashed, and they only have paper records of the customer's account balance for the past 5 weeks. These records give that the balance in the account at the end of each of the past 5 weeks is $810, $840, $870, $900, and $930. Use these records to determine the amount of money that is deposited in the account each week, and determine the balance in the account at the end of week 6.
- While you are running a marathon (26.2 miles), you notice that each mile has a marker. Starting at mile 1, these markers take on the pattern of square, triangle, circle, kite, and then it repeats until the end of the race. If there is an aid station at miles 5, 11, and 20, what are the markers at each of these aid stations?

### Answers:

- If we look at the difference of the balance between consecutive weeks, we see that it goes up by $30 each week ($840 - $810 = $30, $870 - $840 - $30, $900 - $870 = $30, and $930 - $900 = $30). Therefore, the amount that the customer deposits each week is $30. Since the customer deposits $30 each week, and the balance at the end of week 5 is $930, we can find the balance at the end of week 6 by adding $30 to $930 to get $960.
- We can extend the given pattern for 26 terms (the race distance), and then identify what the 5th, 11th, and 20th terms are to find the markers at each of the aid stations in the race. If we abbreviate each shape with their first letter, the mile markers for the whole race are S, T, C, K, S, T, C, K, S, T, C, K, S, T, C, K, S, T, C, K, S, T, C, K, S, T. We see that the 5th term is S, so the aid station at mile 5 has a marker of a square. The 11th term is C, so the aid station at mile 11 has a marker of a circle. The 20th term is K, so the aid station at mile 20 has a marker of a kite.

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