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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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 $3, 6, 12, 24$ next to $3, 6, 9, 12$ . 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 $1, 4, 9, 16$
• 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. $5, 9, 13, 17$ (add $4$ )
Subtracting pattern: the same number is taken away each time. $30, 25, 20, 15$ (subtract $5$ )
Multiplying pattern: each term is multiplied by the same number. $3, 6, 12, 24$ (multiply by $2$ )
Alternating pattern: two steps take turns. $10, 13, 11, 14, 12$ (add $3$ , subtract $2$ )
Growing pattern: the step itself grows. $1, 2, 4, 7, 11$ (add $1$ , then $2$ , then $3$ , then $4$ )
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: $5, 9, 13, 17, \ldots$

Find the change between neighbors

$9 - 5 = 4$ , $13 - 9 = 4$ , $17 - 13 = 4$ → The difference is always $4$

State the rule

Every term is $4$ more than the one before → Rule: start at $5$ , add $4$

Use the rule to continue

$17 + 4 = 21$ , $21 + 4 = 25$ → $21$ and $25$

Answer: The rule is add $4$ . The next two terms are $21$ and $25$ .

Common Mistakes

Choosing a rule after looking only at the first two terms

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

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

Using $10$ jumps to reach the 10th term

Why it's wrong: The first term is already there. From term $1$ to term $10$ there are only $9$ jumps.

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

Thinking every pattern adds the same number

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

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

Forgetting where you are in an alternating pattern

Why it's wrong: In $10, 13, 11, 14, 12, 15$ the next step depends on the last one. Using $+3$ again gives $18$ , which is wrong.

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

Why It Matters

Patterns are everywhere, and spotting them saves time:

Calendars: every Monday of a month is $7$ 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 $3$ 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 $3$ rd, the Mondays are the $3$ rd, $10$ th, $17$ th, $24$ th and $31$ st.

1Try It Yourself

The first Saturday of a month is the $6$ th. Swimming lessons are on every Saturday.

What date is the 4th Saturday of the month?

Step 1: Write the mathematical expression

Start at $6$ and add $3$ jumps of $7$ 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 $2$ tiles uses $6, 8, 10, 12, \ldots$ tiles. The builder can predict any section without drawing it.

2Try It Yourself

A tile design has $5$ tiles in Figure 1, $8$ tiles in Figure 2 and $11$ tiles in Figure 3.

How many tiles are in Figure 6?

Step 1: Write the mathematical expression

Start at $5$ and add $5$ jumps of $3$ :

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 $15$ and add $8$ jumps of $5$ :

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 $9$ 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, 4

the rule could be

+2

\times 2

, but

2, 4, 8

fits only

\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

1

to term

10

you make

9

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

3

shapes.

20 \div 3 = 6

remainder

2

, 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 $4$ or multiply by $2$
Difference: The result of subtracting one term from the next one, for example $9 - 5 = 4$
Alternating pattern: A pattern in which two different steps take turns, such as $+3$ , $-2$ , $+3$ , $-2$
Repeating pattern: A pattern in which the same block of items repeats, such as red, blue, red, blue

Formula Card

Difference check

\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

\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

\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

Teacher Guide: Pattern Recognition

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Calendars

Tiles and Patterns in Building

Saving Money

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Difference check

10th term of an adding pattern

Position in a repeating pattern

More in This Topic

Introduction to Logical Reasoning

Deductive Reasoning

If-Then Statements

Converse and Inverse