Patterns and Rules

🔢 Patterns and Rules

Part 1 of 5 — What Is a Pattern?

Topics in This Part

🔑 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.

💡 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 $+3$
• From $10$ to $13$ is $+3$
• From $13$ to $16$ is $+3$

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!

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: $40, 36, 32, 28, \;?$

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

🔑 Steps to extend any pattern:

1. Find the rule (what changes each step).
2. Apply the rule to the last term.
3. Repeat to get as many new terms as you need.

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) $2, 4, 8, 16, \;?$

🔢 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: $5, 9, 13, 17, 21$

$5 \xrightarrow{+4} 9 \xrightarrow{+4} 13 \xrightarrow{+4} 17 \xrightarrow{+4} 21$

The common difference is +4.

💡 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.

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

🔢 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: $1, 3, 9, 27, 81$

$1 \xrightarrow{\times 3} 3 \xrightarrow{\times 3} 9 \xrightarrow{\times 3} 27 \xrightarrow{\times 3} 81$

The rule is multiply by 3.

⚠️ 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)

🔢 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:

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.

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."

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 $10 \;\rightarrow\; ?$

🔢 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.

🔑 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

⚠️ 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.