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Teacher Guide: Pattern Recognition

Learn to spot the rule behind number and shape patterns, check it against every term, and predict what comes next.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Logical Reasoning. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify the rule of adding, subtracting, multiplying, alternating and growing number patterns
  • Continue a pattern by predicting the next terms
  • Find the 10th term of an adding pattern without listing every term
  • Check that a proposed rule works for every term of a pattern
  • Apply patterns to calendars, tile designs and saving money
Prerequisites
  • • Introduction to Logical Reasoning: the difference between inductive and deductive reasoning
  • • Deductive Reasoning: applying a general rule to a specific case
  • • Adding, subtracting and multiplying whole numbers
  • • Division with remainder
Discussion Starters
  • 1. Where do you see patterns in your day, at home or at school?
  • 2. If someone tells you 2, 4, ... and asks for the next number, how many different answers could be right?
  • 3. Why is it not enough to check a rule on the first two terms?
  • 4. How could you find the 100th term of 3, 7, 11, 15 without writing all the terms?
  • 5. Can you invent a pattern that is hard for a friend to guess? What makes it hard?
Common Misconceptions

Every number pattern is an adding pattern

Remediation: Show 3,6,12,243, 6, 12, 24 next to 3,6,9,123, 6, 9, 12. Let students compute the differences for both and see that only the second one has equal differences.

The nth term is the first term plus n steps

Remediation: Use a ladder or a row of chairs: walking from chair 1 to chair 10 takes 9 steps. Check the method on a term you already know, like the 3rd term.

A rule that fits the first two terms is proven

Remediation: Play Guess My Rule with 1, 2, ... and reveal 4 (doubling) or 3 (adding 1). Students learn to ask for more terms and to test every pair.

Differentiation Ideas

For Struggling Students:

  • • Start with adding patterns only and write the differences in a different color
  • • Use counters or tiles to build each figure of a tile pattern
  • • Give the rule and ask only for the next term before asking for the rule

For On-Level Students:

  • • Mix adding, multiplying and alternating patterns in one worksheet
  • • Find the 10th term of adding patterns and check it on a known term
  • • Solve calendar and savings problems with patterns

For Advanced Students:

  • • Find the 50th or 100th term of an adding pattern
  • • Work with patterns whose differences grow, such as square numbers 1,4,9,161, 4, 9, 16
  • • Create their own pattern puzzles with a hidden rule for classmates
Standards Alignment
  • 4.OA.C.5 (CCSS.MATH.CONTENT.4.OA.C.5)

    Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself

  • 5.OA.B.3 (CCSS.MATH.CONTENT.5.OA.B.3)

    Generate two numerical patterns using two given rules and identify relationships between corresponding terms

  • MP.7 (CCSS.MATH.PRACTICE.MP7)

    Look for and make use of structure

  • MP.8 (CCSS.MATH.PRACTICE.MP8)

    Look for and express regularity in repeated reasoning

Lesson Resources
  • visualSequence Visualizer

    Change the first term and the step and watch the terms and bars grow

  • activityGuess My Rule

    One student secretly picks a rule and says terms one by one; the class guesses the rule and tests it on the next term

  • worksheetPattern Detective

    Mixed patterns where students write the differences, name the rule and find the 10th term

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A pattern is a list of numbers or shapes that follows a rule. Each item in the list is called a term.
To recognize a pattern, look at how each term changes into the next one:
  • Adding pattern: the same number is added each time. 5,9,13,175, 9, 13, 17 (add 44)
  • Subtracting pattern: the same number is taken away each time. 30,25,20,1530, 25, 20, 15 (subtract 55)
  • Multiplying pattern: each term is multiplied by the same number. 3,6,12,243, 6, 12, 24 (multiply by 22)
  • Alternating pattern: two steps take turns. 10,13,11,14,1210, 13, 11, 14, 12 (add 33, subtract 22)
  • Growing pattern: the step itself grows. 1,2,4,7,111, 2, 4, 7, 11 (add 11, then 22, then 33, then 44)
  • Repeating pattern: a block repeats again and again, like circle, square, triangle, circle, square, triangle
A rule only counts if it works for every term, not just the first two. Finding a rule from examples is inductive reasoning; checking it on all terms keeps you honest.

Worked Examples

Find the rule and the next two terms: 5,9,13,17,…5, 9, 13, 17, \ldots
1

Find the change between neighbors

9−5=49 - 5 = 4, 13−9=413 - 9 = 4, 17−13=417 - 13 = 4 → The difference is always 44
2

State the rule

Every term is 44 more than the one before → Rule: start at 55, add 44
3

Use the rule to continue

17+4=2117 + 4 = 21, 21+4=2521 + 4 = 25 → 2121 and 2525

Common Mistakes

Choosing a rule after looking only at the first two terms

Why it's wrong: In 2,4,8,162, 4, 8, 16 the first jump is +2+2, but 4+2=64 + 2 = 6, not 88. Many rules fit two numbers.

Correct: Test the rule on every pair of neighbors. Here ×2\times 2 works for all of them: 2,4,8,162, 4, 8, 16.

Using 1010 jumps to reach the 10th term

Why it's wrong: The first term is already there. From term 11 to term 1010 there are only 99 jumps.

Correct: 10th term == first term +9×+ 9 \times step. For 4,7,10,…4, 7, 10, \ldots: 4+9×3=314 + 9 \times 3 = 31, not 4+10×3=344 + 10 \times 3 = 34.

Thinking every pattern adds the same number

Why it's wrong: In 2,6,18,542, 6, 18, 54 the differences are 4,12,364, 12, 36, so it is not an adding pattern.

Correct: If the differences are not equal, try multiplying (×3\times 3 here), alternating steps, or growing differences.

Forgetting where you are in an alternating pattern

Why it's wrong: In 10,13,11,14,12,1510, 13, 11, 14, 12, 15 the next step depends on the last one. Using +3+3 again gives 1818, which is wrong.

Correct: Write the changes under the terms (+3,−2,+3,…+3, -2, +3, \ldots) and continue the turn: 15−2=1315 - 2 = 13.

Why It Matters

Patterns are everywhere, and spotting them saves time:
  • Calendars: every Monday of a month is 77 days after the last one, so you can find any date without counting day by day
  • Tiles and building: if each new row of a floor needs 33 more tiles, you can plan how many tiles to buy
  • Saving money: putting the same amount in a piggy bank each week lets you predict when you reach your goal
  • Music and art: rhythms and borders are repeating patterns
  • Science and computers: scientists look for patterns in data, and computers are trained to recognize patterns in pictures and sounds
Pattern recognition is the first step of logical thinking: you notice a rule, test it, and then use it to predict.

Real World Applications

Calendars

The same weekday comes back every $7$ days, so dates of that weekday form an adding pattern.

Example:

If the first Monday of a month is the 33rd, the Mondays are the 33rd, 1010th, 1717th, 2424th and 3131st.

1Try It Yourself

The first Saturday of a month is the 66th. Swimming lessons are on every Saturday.

What date is the 4th Saturday of the month?

Step 1: Write the mathematical expression

Start at 66 and add 33 jumps of 77 days:

Tiles and Patterns in Building

Builders and designers use tile patterns for floors, paths and walls, and they need to know how many tiles each part uses.

Example:

A path where each new section adds 22 tiles uses 6,8,10,12,…6, 8, 10, 12, \ldots tiles. The builder can predict any section without drawing it.

2Try It Yourself

A tile design has 55 tiles in Figure 1, 88 tiles in Figure 2 and 1111 tiles in Figure 3.

How many tiles are in Figure 6?

Step 1: Write the mathematical expression

Start at 55 and add 55 jumps of 33:

Saving Money

Saving the same amount every week makes the total in a piggy bank grow in an adding pattern.

Example:

With 10 dollars at the start and 4 dollars saved each week, the totals are 14, 18, 22, 26 dollars after weeks 1 to 4.

3Try It Yourself

Nora has 15 dollars in her piggy bank. She puts in 5 dollars every week.

How much money will she have after 8 weeks?

Step 1: Write the mathematical expression

Start at 1515 and add 88 jumps of 55:

Key Takeaways

  • 1A pattern follows a rule, and each item in it is called a term
  • 2First look at the differences between neighbors: equal differences mean an adding or subtracting pattern
  • 3If the differences are not equal, try multiplying, alternating steps or growing differences
  • 4A rule is only correct if it works for every term, not just the first two
  • 5To get the 10th term of an adding pattern, add 99 steps to the first term
  • 6In a repeating pattern, divide the position by the length of the block and use the remainder

Frequently Asked Questions

How do I start when I see a new pattern?

Write the difference between each pair of neighboring terms under the list. If they are all the same, you have an adding or subtracting pattern. If not, try multiplying or look for a pattern in the differences.

Can a pattern have more than one rule?

With only two or three terms, yes, several rules can fit. That is why you need enough terms and must check your rule on all of them. With 2,42, 4 the rule could be +2+2 or ×2\times 2, but 2,4,82, 4, 8 fits only ×2\times 2.

Why do I add 9 steps and not 10 to get the 10th term?

The first term is already in place. To go from term 11 to term 1010 you make 99 jumps, just like walking from the 1st to the 10th step of a ladder.

How do I find the 20th shape in a repeating pattern?

Count the shapes in one block. For circle, square, triangle the block has 33 shapes. 20÷3=620 \div 3 = 6 remainder 22, so the 20th shape is the 2nd shape of the block: a square.

Glossary

Pattern
A list of numbers or shapes that follows a rule
Term
One item in a pattern; the 1st term is the first item, the 2nd term the next one, and so on
Rule
The instruction that tells you how to get from one term to the next, such as add 44 or multiply by 22
Difference
The result of subtracting one term from the next one, for example 9−5=49 - 5 = 4
Alternating pattern
A pattern in which two different steps take turns, such as +3+3, −2-2, +3+3, −2-2
Repeating pattern
A pattern in which the same block of items repeats, such as red, blue, red, blue

Formula Card

Difference check

difference=next term−term\text{difference} = \text{next term} - \text{term}

If all differences are equal, the pattern adds (or subtracts) that number

10th term of an adding pattern

10th term=1st term+9×step\text{10th term} = \text{1st term} + 9 \times \text{step}

From the 1st to the 10th term there are 9 jumps

Position in a repeating pattern

position÷block length=… remainder r\text{position} \div \text{block length} = \ldots \text{ remainder } r

The remainder $r$ tells you which item of the block it is; remainder $0$ means the last item

More in This Topic

Introduction to Logical Reasoning

Intermediate25 min

Deductive Reasoning

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If-Then Statements

Intermediate25 min

Converse and Inverse

Intermediate25 min