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Extending Patterns: Continue the Pattern Step by Step

This lesson explains how to extend patterns by first uncovering the rule that governs a sequence, then using that rule to continue the pattern with confidence. It covers repeating patterns made of shapes or colors as well as growing number patterns, with worked examples showing how to predict the next several terms.

What Does It Mean to Extend a Pattern?

Extending a pattern means figuring out what comes next in a sequence, and then continuing to add more terms after that. Before you can extend a pattern, you first need to understand it. This is closely tied to identifying patterns, since you cannot continue a pattern until you know the rule that makes it tick. Once the rule is clear, extending the pattern is simply a matter of applying that same rule again and again.

Patterns generally fall into two types. A repeating pattern cycles through the same small set of items over and over, like circle, square, circle, square. A growing pattern changes by a consistent amount each time, like \( 2, 5, 8, 11, \ldots \), where each term is 3 more than the one before it.

Steps to Continue the Pattern

No matter what kind of pattern you are working with, the same three steps work every time.

Look closely at the terms you already have and describe what changes from one term to the next.
Write the rule in your own words, such as "add 4 each time" or "the shapes repeat every 3 items."
Apply the rule to the last known term to find the next one, then repeat the rule again to keep extending.

Extending a Repeating Pattern

In a repeating pattern, the same block of items keeps cycling. To extend it, find the size of the repeating block and figure out where in the cycle you left off. Look at the pattern below.

The block "circle, square" repeats, so the missing shape must be a square.

The repeating block here is "circle, square," which has 2 items. Since the pattern always alternates, the shape after the fifth circle must be a square, matching the same two-item cycle. You could keep extending this pattern indefinitely just by repeating "circle, square" as many times as needed.

Extending a Growing Number Pattern

Growing patterns change by a constant amount, called the common difference, each time you move to the next term. If the first term is \( a_1 \) and the common difference is \( d \), then the value of any term \( n \) can be found with \( a_n = a_1 + (n-1)d \).

Consider the pattern \( 5, 8, 11, 14, \ldots \). Each term is 3 more than the one before, so \( d = 3 \). To extend it, keep adding 3.

Using the rule, the fifth term is \( 14 + 3 = 17 \), and the sixth term is \( 17 + 3 = 20 \). You can check this with the formula too: \( a_6 = 5 + (6-1)(3) = 5 + 15 = 20 \), which matches. Growing patterns like this one are explored in more depth in creating growing patterns, which focuses on building these sequences from scratch.

Worked Example

Extend the pattern \( 2, 6, 10, 14, \ldots \) by three more terms.

Find the change between terms: \( 6 - 2 = 4 \), \( 10 - 6 = 4 \), \( 14 - 10 = 4 \). The common difference is \( d = 4 \).
Add 4 to the last known term repeatedly: \( 14 + 4 = 18 \), \( 18 + 4 = 22 \), \( 22 + 4 = 26 \).
The extended pattern is \( 2, 6, 10, 14, 18, 22, 26 \).

Once you are comfortable extending patterns like this, it is useful to represent them in other ways, such as a table of values or a graph. This connects directly to translating patterns, where the same rule is shown using a different format.

Common Mistakes to Watch For

A common error is assuming a pattern repeats when it actually grows, or the opposite. Always check at least three or four terms before deciding on a rule, since a single pair of terms is not enough to be sure. Another mistake is applying the rule only once instead of repeating it for every new term you add. Always double check your extended terms by working backward, subtracting or reversing the rule to see if you return to the term before it.

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