Number Patterns with Subtraction Rules

A subtraction rule tells you exactly how much to subtract to get from one term in a pattern to the next term. For example, look at this sequence:

\( 100, \; 92, \; 84, \; 76, \; 68 \)

Each number is 8 less than the one before it. The rule for this pattern is "subtract 8," which can also be written as \( -8 \). Because the same amount is subtracted every time, the pattern is predictable, and you can use the rule to continue it or to check whether a number belongs in it.

To find the rule hidden inside a decreasing number pattern, subtract each term from the term right before it, and check that you get the same result every time.

1. Pick two consecutive terms, for example 92 and 84.
2. Subtract the second from the first: \( 92 - 84 = 8 \).
3. Check another pair, such as 84 and 76: \( 84 - 76 = 8 \).
4. Since both differences are 8, the rule is "subtract 8," or \( -8 \).

If the difference changes from one pair to the next, the pattern is not a simple constant subtraction rule, and you would need a different strategy to describe it. Working comfortably with borrowing and place value, as covered in Subtraction with Tens and Ones, makes these difference checks much faster.

Once you know the rule, you can extend the pattern as far as you like by repeating the subtraction. Continuing the pattern \( 100, 92, 84, 76, 68, \ldots \) with the rule \( -8 \):

\( 68 - 8 = 60 \)

\( 60 - 8 = 52 \)

So the next two terms are 60 and 52. The pattern continues as \( 100, 92, 84, 76, 68, 60, 52, \ldots \)

Sometimes a term is missing from the middle or the end of a pattern, and you need to use the rule to fill it in. Consider this pattern with a gap:

\( 50, \; 43, \; \_\_, \; 29, \; 22 \)

First find the rule from terms you can see. Since \( 50 - 43 = 7 \) and \( 29 - 22 = 7 \), the rule is \( -7 \). Apply it to the term before the gap: \( 43 - 7 = 36 \). The missing term is 36, and the full pattern is \( 50, 43, 36, 29, 22 \). This kind of reasoning is closely related to Identifying the Missing Digits (Subtraction), where a missing digit inside a single subtraction problem is found instead of a missing term in a sequence.

Because subtraction and addition undo each other, you can also work backward through a decreasing pattern by adding. If the rule going forward is \( -7 \), then going backward the rule is \( +7 \). This is useful for checking your work: starting from 22 and adding 7 repeatedly should rebuild the same pattern in reverse, \( 22, 29, 36, 43, 50 \).

A number pattern can either increase or decrease. In an increasing pattern, the rule adds the same amount each time, such as \( +5 \). In a decreasing pattern, the rule subtracts the same amount each time, such as \( -5 \). The first step in any number pattern problem is to decide which type you are looking at: check whether the numbers are getting bigger or smaller, and that tells you whether to look for an addition rule or a subtraction rule.

Estimating first can help you catch mistakes before you commit to a full subtraction, especially with larger numbers; the strategies in Subtraction using Estimation are useful here. It also helps to write each difference above the arrow between terms, the way it is shown in the diagrams above, so you can visually confirm the rule stays the same all the way through the pattern before you use it to extend or complete it.

Related lessons

• Math 3