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Creating Growing Patterns

This lesson shows how to create growing patterns by choosing a starting term and a consistent rule, such as adding or multiplying by the same amount each step. Students practice building both shape patterns and number sequences, checking that every step follows the rule.

Introduction

A growing pattern is a sequence of numbers or shapes that changes by a consistent rule from one step to the next, getting larger (or smaller) each time. Before you can create a growing pattern, it helps to know how to spot one; see identifying patterns for a refresher on recognizing the difference between a growing pattern and a repeating one.

In this lesson, you will learn how to build your own growing patterns from scratch, using both shapes and numbers, by choosing a starting point and a rule that stays the same at every step.

What makes a pattern "growing"

A repeating pattern cycles through the same small set of items over and over, like circle, square, circle, square. A growing pattern never repeats the same value twice in that way. Instead, each term is produced by applying the same operation to the term before it. That operation might be:

Adding the same number each time (for example, add \( 4 \))
Multiplying by the same number each time (for example, multiply by \( 2 \))
Adding one more shape or piece at every stage

Because the rule is consistent, once you know the starting term and the rule, you can create as many new terms as you want.

Steps for creating a growing pattern

Pick a starting term. This is stage \( 1 \) of your pattern, for example the number \( 1 \) or a single shape.
Choose a consistent rule. Decide exactly how the pattern will change from one step to the next, such as add \( 3 \) or double the previous term.
Apply the rule repeatedly. Use the same operation on each new term to generate the next one.
Check your work. Look back over the terms you created and confirm the same change happens between every consecutive pair.

Creating a growing pattern with shapes

Suppose you decide the rule is "add one more square than the stage before, stacked in a column." Starting with \( 1 \) square at stage \( 1 \), the pattern grows like this:

Each stage adds one more square than the previous stage: 1, 2, 3, 4 squares.

The rule here is simple to state in words: each stage has one more square than the last. This kind of visual pattern is a good way to introduce growing patterns before moving to pure number sequences.

Creating a growing pattern with numbers

Now build a number pattern using a starting term and a rule of "add \( 3 \) to the previous term." Starting at \( 1 \):

\( 1, \ 4, \ 7, \ 10, \ 13, \ ... \)

Each term can also be written using a rule based on the step number \( n \). Since the first term is \( 1 \) and every step adds \( 3 \), the rule is \( a_n = 3n - 2 \). Checking this: at \( n = 1 \), \( a_1 = 3(1) - 2 = 1 \); at \( n = 4 \), \( a_4 = 3(4) - 2 = 10 \), which matches the list above.

Worked example

Create a growing pattern that starts at \( 5 \) and doubles at every step.

Stage 1: \( 5 \)
Stage 2: \( 5 \times 2 = 10 \)
Stage 3: \( 10 \times 2 = 20 \)
Stage 4: \( 20 \times 2 = 40 \)

The pattern is \( 5, \ 10, \ 20, \ 40, \ ... \) In this pattern, the value multiplies by \( 2 \) each time rather than adding a fixed amount, which shows that a growing pattern's rule does not have to be addition.

Common mistakes to avoid

Changing the rule partway through, for example adding \( 2 \) once and then adding \( 3 \) the next time.
Forgetting to fix the starting term before applying the rule, which makes every later term shift.
Mixing up a growing pattern with a repeating pattern, where the same set of values cycles instead of changing.

Once you can create a growing pattern confidently, practice continuing one that someone else started; see extending patterns for more on that next step.

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Math 1