Read

Number Patterns with Adding Rules

This lesson explains number patterns with adding rules: how to spot the constant amount added between terms, write the rule in words, and extend a sequence to find missing or next numbers, with worked examples and real-world practice.

Introduction

A number pattern with an adding rule is a list of numbers that grows in a predictable way because the same amount is added each time you move from one number to the next. Learning to find that hidden "adding rule" is one of the most useful number sense skills you can build, because it helps you extend sequences, fill in missing numbers, and spot patterns in everyday situations like counting money or measuring growth.

What Is a Number Pattern with an Adding Rule?

Look at this sequence of numbers:

\( 3, \ 7, \ 11, \ 15, \ 19 \)

Each number is bigger than the one before it, and the amount it grows by never changes. To check, subtract each number from the one that comes right after it:

\( 7 - 3 = 4 \), \( 11 - 7 = 4 \), \( 15 - 11 = 4 \), \( 19 - 15 = 4 \)

Since the difference is always 4, the pattern's adding rule is "start at 3 and add 4 each time." This kind of pattern is sometimes called a growing pattern, because the numbers keep getting bigger by the same fixed amount.

A number pattern chain: each term is found by adding 4 to the term before it.

How to Find the Adding Rule in a Pattern

To find the adding rule hiding inside any number sequence, follow these steps:

1. Pick two numbers that sit right next to each other in the pattern.
2. Subtract the smaller number from the larger number to find the difference.
3. Repeat this for a few more pairs in the list.
4. If the difference is always the same number, that number is the adding rule.

For example, in the pattern \( 6, \ 12, \ 18, \ 24 \), subtracting gives \( 12 - 6 = 6 \), \( 18 - 12 = 6 \), and \( 24 - 18 = 6 \), so the rule is add 6. Practicing basic subtraction and addition facts, such as those covered in Addition with Tens and Ones, makes it much faster to spot these differences.

Extending a Number Pattern

Once you know the adding rule, you can continue the pattern as far as you like by adding the rule amount to the last known term. Take the pattern from before, \( 3, \ 7, \ 11, \ 15, \ 19 \), where the rule is add 4. To find the next terms:

\( 19 + 4 = 23 \), \( 23 + 4 = 27 \), \( 27 + 4 = 31 \)

So the extended pattern becomes \( 3, \ 7, \ 11, \ 15, \ 19, \ 23, \ 27, \ 31 \). Notice that each new number simply reuses the same rule, over and over, without changing.

Extending a pattern: the last known term plus the adding rule gives the next term.

Worked Example: A Real-World Adding Rule

Suppose Mia saves money every week and her savings follow this pattern:

Week 1: \( 15 \) dollars, Week 2: \( 30 \) dollars, Week 3: \( 45 \) dollars, Week 4: \( 60 \) dollars

Subtracting consecutive terms shows \( 30 - 15 = 15 \), \( 45 - 30 = 15 \), and \( 60 - 45 = 15 \), so the adding rule is add 15 each week. To predict Week 5's savings, add the rule to the last term: \( 60 + 15 = 75 \) dollars. When the numbers in a pattern get larger, strategies from Adding with Digits Up to 1000 and Addition using Mental Strategies help you add quickly and accurately without mistakes.

Writing the Rule in Words and Numbers

It helps to describe an adding rule in a full sentence, not just as a single number. For the savings pattern above, you could write: "Start at 15 and add 15 each time." Writing the rule this way makes it easy to explain your reasoning and to check whether a new number belongs in the pattern. For instance, does 90 belong in Mia's savings pattern? Since \( 75 + 15 = 90 \), yes, 90 is the next correct term.

Common Mistakes to Avoid

Watch out for these common errors when working with number pattern worksheets:

• Checking only one pair of numbers and assuming the rule is correct without confirming it across the whole list.
• Adding the rule to the wrong term, instead of always adding it to the most recent number in the sequence.
• Mixing up an adding pattern with a pattern that uses a different operation, such as multiplying or repeating a shape instead of a number.

Always test the rule against at least three pairs of numbers before trusting it, and double-check your addition, especially with larger totals.

Practice Makes the Rule Click

The more number patterns you work through, the faster you will get at spotting the adding rule at a glance. Try writing your own sequences, swapping the starting number and the rule amount, and asking a friend to find the pattern you created. This kind of practice builds the same number sense skills used across addition topics, so strong pattern recognition will support your progress everywhere else in math too.

Related lessons

Math 3