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Creating Number Patterns

This Math 3 lesson shows how to create patterns by choosing a starting term and a rule, then applying that rule repeatedly to build repeating and growing sequences of numbers or shapes, with worked examples and common mistakes to avoid.

What Does It Mean to Create a Pattern?

Creating a pattern means choosing a starting point and a rule, and then using that rule over and over to produce a sequence of numbers or shapes. Every pattern you build needs two ingredients: a first term (or a first shape) and a consistent rule that tells you exactly how to get from one term to the next. Once those two pieces are in place, the rest of the pattern is just repeated application of the rule.

This skill sits right alongside identifying patterns, where you look at a finished sequence and figure out the hidden rule. Creating patterns works in the opposite direction: you decide on the rule first, then generate the terms.

Steps to Create a Number Pattern

Follow the same three steps every time you build a number pattern.

Pick a starting number.
Pick an operation and a value for the rule, such as "add \( 4 \)" or "multiply by \( 2 \)".
Apply the rule to the last term you wrote to get the next term, and repeat until you have enough terms.

For an addition rule, the pattern can be written as \( a_n = a_{n-1} + 4 \), which just says each new term equals the term before it plus \(4\).

Worked Example: Building an Addition Pattern

Start at \( 3 \) and use the rule "add \(4\)."

\( 3, \; 3+4=7, \; 7+4=11, \; 11+4=15, \; 15+4=19 \)

The finished pattern is \( 3, 7, 11, 15, 19, \ldots \)

Each box is found by adding 4 to the box before it, starting from 3.

Creating Repeating Patterns

A repeating pattern uses a short block of terms and cycles through that same block again and again. To create one, choose a small unit, such as circle, square, triangle, and repeat it in the same order without changing it. Repeating patterns do not grow larger, they only reuse the same block, which makes them different from the growing patterns described below.

For a shape example, decide the unit is "square, square, circle," then repeat it: square, square, circle, square, square, circle, square, square, circle. The rule here is simply "repeat this block of three."

Creating Growing Patterns

A growing pattern changes size or value at every step, usually because a term or a shape count increases according to the rule. A classic example builds a triangular arrangement of dots, where each row has one more dot than the row before it.

Each new row adds exactly one more circle than the row before it.

The rule for this pattern is "add one more dot than the previous row," which produces the growing sequence \( 1, 2, 3, 4, \ldots \) of row sizes. This kind of pattern is explored in more depth in growing patterns, where the focus is on sequences that increase by a changing amount at each step.

Creating Patterns with Multiplication or Subtraction Rules

Not every rule has to be addition. A pattern can also be created using multiplication, such as starting at \( 2 \) and multiplying by \( 3 \) each time to get \( 2, 6, 18, 54, \ldots \), or using subtraction, such as starting at \( 30 \) and subtracting \( 5 \) each time to get \( 30, 25, 20, 15, \ldots \). The steps stay the same: pick the start, pick the operation and value, then apply it repeatedly.

Checking a Pattern You Created

After building a pattern, check it by working through the terms again with the same rule to make sure every gap matches. If the rule was "add \( 4 \)," subtracting \( 4 \) from each term should always give the term right before it. This check is the same skill used when extending patterns, since extending a pattern correctly depends on applying the exact rule that created it in the first place.

Common Mistakes to Avoid

Switching the rule partway through, for example adding \( 4 \) once and then adding \( 5 \) later without meaning to.
Forgetting to state the rule clearly before building the pattern, which makes it hard to check later.
Mixing up a repeating pattern with a growing pattern when the size of the shapes or numbers should stay constant.

Related lessons

Math 3