Interpreting and Extending Patterns

A simple three-step plan

Almost every pattern question can be cracked with the same three steps. Keep them in your back pocket:

• 1. Find the step. Look at how you get from one term to the next. Did the numbers go up or down? By how much?
• 2. Name the rule. Decide what the pattern does: add a number, subtract a number, or multiply by a number, the same way every time.
• 3. Apply the rule. Use it to extend the list or to fill in the missing term.

Worked example #1: 4, 7, 10, 13, __, __

Find the rule, then extend the pattern by two more terms.

• Find the step: 7 − 4 = 3, 10 − 7 = 3, 13 − 10 = 3. The gap is always 3.
• Name the rule: add 3 each time.
• Apply it: 13 + 3 = 16, then 16 + 3 = 19.

  4    7    10   13   __   __
   +3   +3   +3   +3   +3

  4    7    10   13   16   19

Worked example #2: 50, 44, 38, 32, __

Patterns can shrink, too. Watch for numbers going down.

• Find the step: 50 − 44 = 6, 44 − 38 = 6, 38 − 32 = 6. Each term drops by 6.
• Name the rule: subtract 6 each time.
• Apply it: 32 − 6 = 26.

  50   44   38   32   __
   −6   −6   −6   −6

  50   44   38   32   26

Worked example #3: 3, 6, 12, 24, __

Sometimes the gaps keep getting bigger. That is a clue the rule is multiplication, not addition.

• Find the step: the gaps are +3, +6, +12, not constant, so it is not a simple "add." But 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. Each term is double the one before.
• Name the rule: multiply by 2 each time.
• Apply it: 24 × 2 = 48.

  3    6    12   24   __
   ×2   ×2   ×2   ×2

  3    6    12   24   48

Worked example #4: complete the pattern 5, 9, __, 17, 21

Some questions hide the blank in the middle instead of at the end. The plan still works, find the rule from the terms you can see, then fill the gap.

• Find the step: 9 − 5 = 4, and later 21 − 17 = 4. The rule is add 4.
• Name the rule: add 4 each time.
• Apply it: the missing term comes after 9, so 9 + 4 = 13. Check: 13 + 4 = 17, which matches the next term given.

  5    9    __   17   21
   +4   +4   +4   +4

  5    9    13   17   21

Visual patterns: count, then find the rule

Patterns are not always written as numbers. Sometimes you get a row of shapes that grows step by step. The trick is the same: count the dots in each step to turn the picture into numbers, then find the rule.

Counting the dots turns the picture into a number pattern: 1, 3, 5, 7. Now find the rule the usual way.

• Find the step: 3 − 1 = 2, 5 − 3 = 2, 7 − 5 = 2. Each step adds 2 dots (shown in green above).
• Name the rule: add 2 dots each step.
• Apply it: Step 5 would have 7 + 2 = 9 dots.

  Step:   1    2    3    4    5
  Dots:   1    3    5    7    9
           +2   +2   +2   +2

Tips for cracking any pattern

• Check the differences first. Subtract each term from the next. If the difference is the same every time, the rule is add or subtract that number.
• If the gap keeps growing, test multiplication. Divide a term by the one before it. If you get the same answer each time, the rule is "multiply by" that number.
• Watch the direction. Going up means add or multiply; going down means subtract (or, less often, divide).
• Always verify with a second gap. One difference could be a coincidence. Confirm the rule works between at least two pairs of terms before you trust it.
• Turn pictures into numbers. For visual patterns, count the shapes in each step and write the counts as a list.

Your turn: practice problems

Find the rule for each one, then give the missing term(s). Write out the differences so you can see the rule. No peeking until you have tried.

1. Extend two more terms: 6, 11, 16, 21, __, __
2. Find the next term: 80, 72, 64, 56, __
3. Find the next term: 2, 6, 18, 54, __
4. Complete the pattern: 14, __, 26, 32, 38