Patterns and Sequences

Grade 3 · Math · Study Guide

Learn to identify, extend, and create number and shape patterns.

A pattern is something that repeats or grows in a predictable way. Patterns are everywhere in math — in multiplication tables, in shapes, in nature, and in everyday life.

A repeating pattern uses a core that keeps coming back, like: circle, square, triangle, circle, square, triangle... The core here is circle-square-triangle. To continue the pattern, you just repeat the core.

A growing pattern increases (or decreases) by the same amount each time. For example: 3, 6, 9, 12, 15 — this grows by adding 3 each time. The rule is "add 3." You can also have patterns that multiply: 2, 4, 8, 16 — each term is doubled.

Finding the rule is the key skill. Look at how the pattern changes from one term to the next. Is it adding, subtracting, multiplying? Once you know the rule, you can predict any term in the sequence — even the 100th!

Key terms

Pattern

A sequence that follows a predictable rule.

Repeating pattern

A pattern where a core unit keeps cycling. Ex: AB, AB, AB...

Growing pattern

A pattern that increases or decreases by a consistent rule.

Rule

The mathematical operation that describes how a pattern changes.

Term

One number or shape in a sequence.

Flashcards

Repeating pattern

A core unit that cycles. Ex: red, blue, red, blue...

Growing pattern rule: +5

Start with any number and add 5 each time: 5, 10, 15, 20...

Pattern rule: ×2

Double each term: 1, 2, 4, 8, 16...

Shrinking pattern rule: −3

Subtract 3 each time: 30, 27, 24, 21...

Term

One item in a sequence.

Finding a pattern rule

Check the difference or ratio between consecutive terms.

Practice questions

1. What is the next number in the pattern: 5, 10, 15, 20, ___?

   1. 22
   2. 24
   3. 25
   4. 30
   Show answer

   Answer: 25

   Rule: add 5. 20 + 5 = 25.

2. What is the rule for this pattern: 2, 4, 8, 16?

   1. Add 2
   2. Add 4
   3. Multiply by 2
   4. Multiply by 4
   Show answer

   Answer: Multiply by 2

   2×2=4, 4×2=8, 8×2=16. Each term is doubled.

3. The pattern 100, 90, 80, 70 has a rule of "subtract 10."

   Show answer

   Answer: True

   100−10=90, 90−10=80, 80−10=70. Rule is subtract 10.

4. The next term in 3, 6, 9, 12, ___ is ___

   Show answer

   Answer: 15

   Rule: add 3. 12 + 3 = 15.

5. Which number is missing? 7, 14, ___, 28, 35

   1. 18
   2. 20
   3. 21
   4. 22
   Show answer

   Answer: 21

   Rule: add 7. 14 + 7 = 21. Check: 21 + 7 = 28 ✓.

Fun facts

• Fibonacci numbers (1, 1, 2, 3, 5, 8, 13...) are a growing pattern where each term is the sum of the two before it — and this pattern appears in sunflower seeds, pinecones, and seashells!
• Patterns in multiplication tables have been used by traders for over 2,000 years to do fast calculations without calculators.

Get a personalized course on Patterns and Sequences built around your child.