Number Patterns: Find the Rule, A Grade 6 Guide

A number pattern is a list of numbers that follows a fixed rule. You find the rule by checking the gap between terms and testing it on every step. I guess one operation, test it on the second and third terms, and trust it only if it holds, because a rule that fails one term is not the rule.

What Is a Number Pattern?

A number pattern is a list of numbers that follows a fixed rule, so each term comes from the one before it or from its position in the list. In 3, 6, 9, 12, each step adds 3, and once you name that rule the next terms write themselves without guessing.

A random list has no promise. A pattern makes one: the next term must follow the rule, and you can test any guess against the list. That promise is what makes patterns fun to crack.

Think of stairs with equal steps. Each step rises the same height, so you know the next step before you climb it. A pattern works the same way, and the term is just one number in the list, counted by its position.

Write positions above the terms when you work. Position 1 holds 3, position 2 holds 6, position 3 holds 9, and the table keeps the order honest.

How Do You Find the Rule?

Find the rule by checking the gap between terms, then test if one operation explains every step, and check the rule against a table of values before you trust it. A rule that fits the first two terms is a guess, not a rule, until it survives the third and fourth terms.

Work it in three moves. First, write the gaps between neighbours. Second, guess one operation that fits the gaps, and third, test the guess on every term, not just the first two.

Take 5, 9, 13, 17. The gaps are all 4, so the rule is add 4, and the next term is 21.

One gap alone would not prove it, but four equal gaps do. I ask children to say the gap out loud before they name the rule, because the saying slows the guessing down.

What Is the Position-to-Term Rule?

The position-to-term rule gives any term from its position, without writing out the terms before it, because you plug the position number into the rule. With the rule 3n + 1, the 10th term is 31, since 3 times 10 plus 1 equals 31, and the 100th term is 301 likewise.

Call the position n. In 3n + 1, n is the position and the answer is the term, so position 1 gives 4, position 2 gives 7, and position 3 gives 10. The list runs 4, 7, 10, 13, and the gap stays 3 all the way down.

Writing a rule with a letter is the first step into algebra. I explain letters in plain words in my Algebra in Real Life guide, and the same letter habit carries straight into this work.

This is where position and value meet. The position is the independent quantity and the term value is the dependent quantity, and that link is exactly what Common Core standard 6.EE.C.9 asks students to write as an equation and read from a table.

What About Two-Step Rules?

A two-step rule needs two operations per term, like multiply by 3 then take away 1, and you must test the guess on the second and third terms, not the first. One term can flatter a wrong rule, so a rule earns trust only when it builds every term in the list.

Take 2, 5, 14, 41. Guess multiply by 3 then take away 1, and test it: 2 times 3 minus 1 is 5, 5 times 3 minus 1 is 14, and 14 times 3 minus 1 is 41. The guess holds three times, so it earns trust.

Test the same guess on a wrong list and it breaks fast. That break is good news, because it tells you to guess again instead of marching on with a wrong rule. I praise a caught mistake in class, because catching it is the skill.

Can You Build Your Own Pattern From a Rule?

You build a pattern from a rule by feeding positions into the rule and writing each answer as a term, then checking the list against a table of values. Start with position 1, work the rule in order, and a wrong step shows up because tables keep the positions honest.

Build from 2n + 3. Position 1 gives 5, position 2 gives 7, position 3 gives 9, and position 4 gives 11. The list reads 5, 7, 9, 11, and each gap is 2, which matches the rule.

Now build your own. Pick a rule, write four terms, and hand the list to a friend with the rule covered. If she finds your rule from the terms alone, you built it clean.

Where Do Patterns Show Up in Real Life?

Patterns show up in saving money, because a weekly top-up turns savings into a list with a steady gap and a rule you can trust. Save 5 pounds, add 3 each week, and the totals run 5, 8, 11, 14, with the rule 3n + 2 giving week 10 as 32 pounds.

Step 1: You save 5 pounds in week 1 and add 3 pounds each week after. The totals run 5, 8, 11, 14, and the gap stays 3 all the way down the list.

Step 2: Write the rule from the positions. Week n holds 3n + 2 pounds, because week 1 gives 5 and week 2 gives 8. Check week 3: 3 times 3 plus 2 is 11, and the list agrees.

Step 3: Ask for a far week without listing. Week 10 holds 3 times 10 plus 2, which is 32 pounds. No long list, no counting on fingers, just the rule doing the work.

Try It Yourself: 3 Practice Questions

Cover the answers, try all three, then check. For a full page of the same practice, print my free Number Patterns Worksheet for Grade 6.

1. Extend this sequence and name the rule: 7, 12, 17, 22, __.
2. Find the missing term: 4, 9, __, 19, 24.
3. The rule is 5n – 2. What is the 10th term?

Answers

1. 27. The gap is 5, so the rule is add 5, and 22 + 5 = 27.
2. 14. The gap is 5 all through, so 9 + 5 = 14, and 14 + 5 = 19.
3. 48. Put n = 10 into the rule: 5 times 10 minus 2 is 48.

Key Takeaways

• Gaps first: write the gaps before you guess a rule.
• Test three terms: a rule must survive the second and third terms, not just the first.
• Position-to-term: plug the position into the rule for far terms like the 10th or 100th.
• Two steps: test both operations on later terms before you trust the guess.
• Build and check: write a table, and let the table catch a wrong step.

If your child found all three answers, she is ready for a longer page. For more free practice at this level, browse our Grade 6 math worksheets page. Every sheet is free to print.