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Word Problems Involving Patterns

This lesson teaches students how to read a word problem, spot the repeating rule in a number pattern, and use that rule to find missing or future values, with worked examples of growing and shrinking patterns.

What Are Word Problems Involving Patterns?

A word problem involving patterns describes a real-life situation where numbers change in a predictable way. Instead of just being handed a list of numbers, you have to read a short story, pull out the numbers that are given, and figure out the rule that connects them. Once you know the rule, you can use it to answer a question about a term you were not given directly, like "how many stickers will she have after 6 days?"

These problems build directly on skipcounting and simple addition or subtraction rules. If you have not reviewed those ideas yet, it helps to look back at adding with patterns up to 100 and subtracting with patterns up to 100 before tackling word problems, since the pattern rules are the same, they are just hidden inside a sentence instead of written as a bare list.

Steps for Solving a Pattern Word Problem

Most pattern word problems can be solved by following the same four steps every time.

1. Read the problem carefully and write down the numbers you are given, in order.
2. Find the rule by comparing each number to the one before it. Ask, "what happened to get from one number to the next?"
3. Use the rule to continue the pattern, either by counting forward step by step or by using addition or subtraction directly.
4. Answer the actual question being asked, making sure your answer matches what the problem wants (a total, a missing term, or a specific step).

Example 1: A Growing Pattern

Suppose a garden has 1 row of 4 flowers, 2 rows of 8 flowers, and 3 rows of 12 flowers. How many flowers are in 5 rows, if the pattern continues?

First, list what you know: row 1 has \( 4 \) flowers, row 2 has \( 8 \) flowers, row 3 has \( 12 \) flowers. Comparing each value to the one before shows the rule: each row adds \( 4 \) more flowers than the last, since \( 8 - 4 = 4 \) and \( 12 - 8 = 4 \).

Organizing this in a table makes the pattern easy to see and extend.

Each row adds 4 flowers, so row 4 has 16 flowers.

Following the rule, row 4 has \( 12 + 4 = 16 \) flowers, and row 5 has \( 16 + 4 = 20 \) flowers. So the garden has \( 20 \) flowers in row 5.

Example 2: A Shrinking Pattern

A tank starts with 30 liters of water. Every hour, 6 liters drain out. How much water is left after 4 hours?

Here the story tells you the rule directly: subtract \( 6 \) each hour. Starting at \( 30 \), the amounts after each hour are \( 30 - 6 = 24 \), then \( 24 - 6 = 18 \), then \( 18 - 6 = 12 \), then \( 12 - 6 = 6 \). After 4 hours, only \( 6 \) liters remain.

Notice that this problem gave the rule in words ("6 liters drain out every hour"), while Example 1 required you to find the rule yourself by comparing the numbers. Both skills matter, since some word problems state the rule and others hide it inside the data.

Example 3: Finding a Missing Term

A number pattern word problem might also ask you to fill in a gap. For example: "On Monday, Jae read 5 pages. On Tuesday, he read 10 pages. On Wednesday, he read 15 pages. If the pattern continues, how many pages will he read on Friday?"

The rule is adding \( 5 \) each day, matching the days as steps: Monday is step 1 with \( 5 \) pages, Tuesday is step 2 with \( 10 \) pages, and so on. Thursday (step 4) would be \( 20 \) pages, and Friday (step 5) would be \( 25 \) pages. Since the number of pages equals \( 5 \) times the day number, you could also jump straight to \( 5 \times 5 = 25 \) without writing out every day.

Tips for Working with Pattern Word Problems

Always check your rule against every number you were given, not just the first two. A rule that works for the first pair of numbers but not the third is not the correct rule. It also helps to decide early whether the pattern is growing (values getting larger, usually through addition or multiplication) or shrinking (values getting smaller, usually through subtraction or division), since that tells you what kind of rule to look for.

Some problems combine more than one type of change in the same story, mixing addition steps with subtraction steps or changing the size of the step partway through. Practicing with mixed patterns is a good way to get comfortable recognizing several kinds of rules at once, which is exactly what more advanced pattern word problems require.

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