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Mixed Number Patterns

This lesson explains mixed patterns, sequences of numbers that combine growing and shrinking steps or switch between addition and subtraction rules. Students learn to compare terms, spot the repeating rule, and extend sequences confidently using tables and worked examples.

Introduction

A number pattern is a list of numbers that follows a rule. In a mixed pattern, that rule is not always the same simple step every time. Sometimes the numbers grow, sometimes they shrink, and sometimes the pattern switches between adding and subtracting within the very same sequence. Learning to spot these mixed rules is an important step after mastering simple patterns such as adding with patterns up to 100 and subtracting with patterns up to 100.

What Are Mixed Patterns?

In a simple pattern, every step uses the same rule. For example, \( 2, 5, 8, 11, 14 \) always adds 3. A mixed pattern uses more than one type of step, or it changes direction partway through. Two common types of mixed patterns are:

Growing patterns: each term is larger than the one before it.
Shrinking patterns: each term is smaller than the one before it.

A mixed pattern might grow for part of the sequence and shrink for another part, or it might alternate between an addition step and a subtraction step every time.

A growing pattern with rule \( +3 \) compared to a shrinking pattern with rule \( -4 \).

Finding the Rule in a Mixed Pattern

To find the rule of any pattern, subtract each term from the one that follows it. Write these differences underneath the sequence. If the differences are all the same, the pattern uses one simple rule. If the differences change, for example alternating between a positive number and a negative number, you have a mixed pattern.

Look at this sequence: \( 4, 7, 5, 8, 6, 9 \). Comparing each pair of terms gives:

\( 4 \to 7 \): \( +3 \)
\( 7 \to 5 \): \( -2 \)
\( 5 \to 8 \): \( +3 \)
\( 8 \to 6 \): \( -2 \)
\( 6 \to 9 \): \( +3 \)

The rule alternates between \( +3 \) and \( -2 \). Once the rule is spotted, the next term follows the same pattern, so after \( 9 \) the next step is \( -2 \), giving \( 7 \).

A table of differences reveals the alternating \( +3, -2 \) rule in a mixed pattern.

Worked Example

Extend the pattern: \( 30, 25, 28, 23, 26, \ldots \)

Check the differences: \( 30 \to 25 \) is \( -5 \); \( 25 \to 28 \) is \( +3 \); \( 28 \to 23 \) is \( -5 \); \( 23 \to 26 \) is \( +3 \). The rule alternates between \( -5 \) and \( +3 \). Since the last step was \( +3 \) (from 23 to 26), the next step should be \( -5 \), so the next term is \( 26 - 5 = 21 \). The term after that adds 3 again, giving \( 21 + 3 = 24 \).

Growing and Shrinking Patterns in Word Problems

Mixed patterns often show up in real situations, such as a savings amount that goes up some weeks and down other weeks, or a plant that grows one week and is trimmed the next. When a pattern is described in words instead of numbers, the first step is always the same: write out the sequence of numbers, then compare each term to find the rule, just as in a purely numeric pattern. For more practice turning a story into a sequence, see word problems involving patterns.

Practice Tips

Always write the differences between consecutive terms before deciding on a rule.
Check whether the pattern is purely growing, purely shrinking, or alternates between the two.
Look for a repeating cycle of steps, such as \( +3, -2, +3, -2, \ldots \), rather than assuming only one step size.
Use a table to keep terms and differences lined up neatly, especially with longer sequences.
Double check your extended terms by working the rule forward from the last known term.

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