🎯⭐ INTERACTIVE LESSON

Patterns and Rules

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Patterns and Rules - Complete Interactive Lesson

Part 1: What Is a Pattern?

🔢 Patterns and Rules

Part 1 of 5 — What Is a Pattern?

Topics in This Part

Section
What Is a Pattern?
Finding the Rule
Extending a Pattern

🔑 Key Idea: A pattern is a list of numbers (or shapes) that follow the same rule over and over. Once you find the rule, you can keep the pattern going forever!

What Is a Pattern?

A pattern is a group of numbers that change in the same way each time. The numbers in a pattern are called terms.

Look at this pattern:

$2,\; 4,\; 6,\; 8,\; 10$

Each term goes up by 2. The "step" between terms is always the same. That steady step is what makes it a pattern.

Pattern	What changes each step	Rule
$5, 10, 15, 20$	goes up by 5	add 5
$20, 18, 16, 14$	goes down by 2	subtract 2

💡 To check a pattern, look at the jump from one term to the next. If the same thing happens every time, you found a pattern!

Concept Check 🎯

Finding the Rule

The rule tells you what to do to get from one term to the next. To find the rule, ask:

❓ "What do I do to this number to get the next one?"

Example: $7, 10, 13, 16$

From $7$ to $10$ is $+3$

Name the Rule 🔽

Choose the rule that makes each pattern.

Extending a Pattern

To extend a pattern means to keep it going. Once you know the rule, just keep using it!

Example: $9, 14, 19, 24, \;?$

The rule is add 5. So the next term is $24 + 5 = 29$ .

Example:

Keep It Going 🧮

Find the rule, then write the next term in each pattern.

1) $8, 11, 14, 17, \;?$ 2) $60, 50, 40, 30, \;?$ 3)

Part 2: Add & Subtract Patterns

🔢 Patterns and Rules

Part 2 of 5 — Add & Subtract Patterns

🔑 The Idea: The most common patterns use addition or subtraction. We call these arithmetic patterns, and the steady step is called the common difference.

The Common Difference

In an add-or-subtract pattern, the same number is added (or subtracted) every step. That number is the common difference.

Example: $5, 9, 13, 17, 21$

$5 \overset{+ 4}{\to} 9 \overset{}{\to}$

Part 3: Multiply & Divide Patterns

🔢 Patterns and Rules

Part 3 of 5 — Multiply & Divide Patterns

🔑 The Idea: Some patterns grow fast by multiplying, or shrink fast by dividing. Instead of taking the same-size step, each term is a few times bigger (or smaller) than the one before.

Multiplying Patterns

In a multiplying pattern, you multiply by the same number each step. These patterns grow quickly!

Example: $1, 3, 9, 27, 81$

$1 \overset{\times 3}{\to} 3 \overset{\times 3}{\to} 9 \overset{}{}$

Part 4: Input-Output Tables

🔢 Patterns and Rules

Part 4 of 5 — Input-Output Tables

🔑 The Idea: A function table (or input-output table) is a machine. You put a number in, the rule changes it, and a new number comes out. The same rule works on every input.

The Rule Machine

Think of a rule like a machine. The input goes in, the rule does its job, and the output comes out.

Here the rule is add 3:

Input	Rule: add 3	Output
1	$1 + 3$	4
2	$2 + 3$

Part 5: Shape Patterns & Mastery Check

🔢 Patterns and Rules

Part 5 of 5 — Shape Patterns & Mastery Check

You can now (1) find a rule, (2) add and subtract patterns, (3) multiply and divide patterns, and (4) use input-output tables. Let's finish with shape patterns and put it all together!

Patterns Made of Shapes

Patterns aren't only numbers — they can be shapes too! A repeating shape pattern uses a unit that repeats over and over.

Look at this pattern:

$\triangle\; \square\; \square\; \triangle\; \square\; \square\; \triangle\; \square\; \square$

The repeating unit is triangle, square, square. It repeats again and again.

From

10

13

+3

From

13

16

+3

The rule is add 3. ✅

Example: $30, 25, 20, 15$

From $30$ to $25$ is $-5$
The rule is subtract 5. ✅

⚠️ Watch out: Always check more than one jump before you decide on the rule. One jump alone might trick you!

The rule is subtract 4. So the next term is $28 - 4 = 24$ .

🔑 Steps to extend any pattern:

Find the rule (what changes each step).

Apply the rule to the last term.

Repeat to get as many new terms as you need.

5 + 4 9 + 4 13 + 4 17 + 4 21

The common difference is +4.

Pattern	Common difference	Rule
$3, 8, 13, 18$	$+5$	add 5
$100, 90, 80, 70$	$-10$	subtract 10
$12, 19, 26, 33$	$+7$	add 7

💡 To find the common difference, subtract any term from the one after it: $9 - 5 = 4$ . ✅

Find the Common Difference 🧮

Write the number that is added or subtracted each step. (If the pattern goes down, write a negative number, like $-3$ .)

1) $4, 10, 16, 22$ 2) $45, 40, 35, 30$ 3) $7, 15, 23, 31$

Filling In Missing Terms

Sometimes a pattern has a blank in the middle. You can still fill it in using the rule!

Example: $6, 11, \underline{\;?\;}, 21, 26$

The rule is add 5. To find the blank, add 5 to the term before it:

$11 + 5 = 16$

✅ Check: $16 + 5 = 21$ , which matches the next term. So the blank is $16$ .

Example: $30, \underline{\;?\;}, 20, 15$

The rule is subtract 5, so the blank is $30 - 5 = 25$ .

🔑 Trick: To fill a blank, you can add the rule to the term before it, OR work backward from the term after it. Both should agree!

Concept Check 🎯

Patterns in Real Life

Patterns show up everywhere! Word problems often hide an add-or-subtract pattern.

Example: Maya saves money. She starts with $10, and each week she adds $5.

Week	Money saved
1	$10
2	$15
3	$20
4	$25

The rule is add 5 each week. So in Week 5 she will have $25 + 5 = \$ 30$.

💡 Reading a word problem? Look for the starting number and the amount it changes each time. Those two facts give you the whole pattern.

Pattern Word Problems 🧮

1) A plant is $12$ cm tall and grows $3$ cm each week. How tall will it be after 1 more week? (in cm)

2) A theater has $20$ seats in row 1, $24$ in row 2, $28$ in row 3. How many seats are in row 4?

3) Sam has $48$ stickers and gives away $6$ each day. How many will he have after 1 day?

1 \times 3 3 \times 3 9 \times 3 27 \times 3 81

The rule is multiply by 3.

Pattern	Rule
$2, 4, 8, 16$	multiply by 2 (doubling)
$1, 5, 25, 125$	multiply by 5
$3, 30, 300, 3000$	multiply by 10

⚠️ Add vs. Multiply: $2, 4, 6, 8$ adds 2 each time. But $2, 4, 8, 16$ multiplies by 2 each time. Check carefully — the first two terms can look the same!

Add or Multiply? 🎯

Dividing Patterns

In a dividing pattern, you divide by the same number each step. These patterns shrink quickly.

Example: $80, 40, 20, 10$

$80 \xrightarrow{\div 2} 40 \xrightarrow{\div 2} 20 \xrightarrow{\div 2} 10$

The rule is divide by 2 (this is the same as cutting it in half).

Example: $1000, 100, 10, 1$ — the rule is divide by 10.

💡 Helpful link: Dividing by 2 is the opposite of multiplying by 2. If a pattern grows by $\times 2$ , the same pattern read backwards shrinks by $\div 2$ .

Multiply or Divide? 🔽

Pick the rule for each fast-changing pattern.

Next Term 🧮

Find the rule, then write the next term.

1) $1, 2, 4, 8, \;?$ (multiplying) 2) $5, 15, 45, \;?$ (multiplying) 3) $81, 27, 9, \;?$ (dividing)

Notice: the same rule ("add 3") is used on every input.

💡 To find an output, take the input and do the rule. To find the rule, look at how each input changes into its output.

Run the Machine 🧮

The rule is multiply by 4. Find each output.

1) Input $2 \;\rightarrow\;$ output $?$ 2) Input $5 \;\rightarrow\;$ output $?$ 3) Input $10 \;\rightarrow\;$ output $?$

Finding the Rule from a Table

Sometimes you get the table and must discover the rule. Compare each input to its output.

Input	Output
3	9
4	12
6	18

Ask: "What turns 3 into 9?" You could add 6 ( $3+6=9$ )... but check the next row: $4 + 6 = 10$ , not 12. ❌

Try multiply by 3: $3 \times 3 = 9$ ✅, $4 \times 3 = 12$ ✅, $6 \times 3 = 18$ ✅.

The rule is multiply by 3.

⚠️ Always test your rule on EVERY row. A rule that works on one row but fails another is the wrong rule.

Concept Check 🎯

Complete the Table 🔽

The rule is add 8. Choose the correct output for each input.

Two-Step Rules (Challenge!)

Some rule machines do two jobs. For example, "multiply by 2, then add 1."

Input	$\times 2$	then $+1$	Output
1	2	$2+1$	3
2	4	$4+1$	5
5	10	$10+1$	11

So input $4$ gives $4 \times 2 = 8$ , then $8 + 1 = 9$ .

🔑 Do the steps in order, left to right. For "multiply by 2, then add 1," always multiply first, then add.

Two-Step Machine 🧮

The rule is multiply by 3, then add 2. Find each output.

1) Input $1 \;\rightarrow\; ?$ 2) Input $4 \;\rightarrow\; ?$ 3) Input $10 \;\rightarrow\; ?$

🔑 Find the unit: Look for the smallest group that repeats. Here it is 3 shapes long: $\triangle\,\square\,\square$ . The 10th shape would start a new unit, so it is a triangle.

Growing shape patterns can also hide a number pattern — like dots that grow $1, 3, 5, 7$ each step.

Concept Check 🎯

Quick Reference

Pattern type	How to spot it	Example
Add	same amount added	$4, 7, 10, 13$ (add 3)
Subtract	same amount taken away	$20, 16, 12, 8$ (subtract 4)
Multiply	grows fast	$2, 6, 18, 54$ (×3)
Divide	shrinks fast	$80, 40, 20, 10$ (÷2)
Table rule	works on every row	input ×4 → output
Shape unit	smallest part that repeats	🔺🟦🟦

⚠️ Two big rules to remember: (1) Check the rule on more than one step. (2) "Add 2" and "multiply by 2" are different — look past the first two terms!

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.