Mathematics: Algebraic Reasoning – Grade 4

When you divide numbers, sometimes you don't get a perfect answer - you get a remainder! 🧮 Learning how to interpret remainders is a crucial skill that helps you solve real-world problems correctly.

A remainder is the amount left over after division. But here's the important part: what you do with the remainder depends on what the problem is asking! Let's explore the four main ways to handle remainders:

1. Add 1 to the Quotient (Round Up)

Sometimes you need to round up to include everyone or everything. For example:

• 30 students going on a field trip need cars that hold 4 people each
• $30 ÷ 4 = 7$ remainder $2$
• You need 8 cars because 2 students still need transportation! 🚗

2. Use Only the Remainder

Sometimes only the leftover amount matters:

• Gerardo has $\$19$ to share equally among 4 friends, giving the rest to his sister
• $19 ÷ 4 = 4$ remainder $3$
• His sister gets $\$3$ (just the remainder) 💰

3. Drop the Remainder

Sometimes you can only use complete groups:

• Alicia has $\$48$ to buy $\$10$ meals for 5 friends
• $48 ÷ 10 = 4$ remainder $8$
• She can buy 4 complete meals (dropping the remainder) 🍕

4. Express the Remainder as a Fraction

Sometimes you need to show the partial amount:

• 7 cupcakes shared equally among 3 people
• $7 ÷ 3 = 2$ remainder $1$
• Each person gets $2\frac{1}{3}$ cupcakes 🧁

Estimation Techniques:

• Use compatible numbers (numbers that work well together): $32 ÷ 8$ instead of $31 ÷ 8$
• Round to nearby multiples of 10: $47 × 6$ becomes $50 × 6 = 300$
• Check if your answer makes sense by comparing to your estimate

Multiplicative Comparisons: These problems compare quantities using "times as many":

• "Sarah has 24 stickers, which is 3 times as many as Tom has"
• This means: $24 = 3 × \text{Tom's stickers}$
• So Tom has $24 ÷ 3 = 8$ stickers

1. Read carefully - What is the problem asking?
2. Identify the operation - Do you need to multiply or divide?
3. Solve the problem - Show your work clearly
4. Interpret the remainder - What does it mean in this situation?
5. Check your answer - Does it make sense?

⚠️ Don't automatically write "r" for every remainder - Think about what the remainder means in the problem!

⚠️ Don't confuse the remainder with additional whole units - A remainder of 2 doesn't mean 2 more buses; it means 2 people who need space in one more bus!

⚠️ Don't forget to check your multiplication facts - Practice your facts up to 12 × 12 for fluency

Fractions are everywhere in your daily life! 🍕 From sharing pizza to measuring ingredients, understanding how to add and subtract fractions with like denominators helps you solve practical problems.

Fractions have like denominators when they have the same bottom number. For example:

• $\frac{2}{8} + \frac{3}{8}$ (both have 8 as denominator)
• $\frac{5}{12} - \frac{3}{12}$ (both have 12 as denominator)

When fractions have like denominators, you can add or subtract them by working with the numerators (top numbers) while keeping the same denominator.

The Rule: Add the numerators, keep the denominator the same.

Example: Sara read $\frac{2}{8}$ of her book on Friday and $\frac{3}{8}$ on Saturday. How much did she read in total?

$\frac{2}{8} + \frac{3}{8} = \frac{2+3}{8} = \frac{5}{8}$

Sara read $\frac{5}{8}$ of her book! 📚

The Rule: Subtract the numerators, keep the denominator the same.

Example: Jose completed $\frac{8}{12}$ of his exercise program with pull-ups. What fraction was NOT pull-ups?

$1 - \frac{8}{12} = \frac{12}{12} - \frac{8}{12} = \frac{4}{12}$

So $\frac{4}{12}$ of the program was not pull-ups! 💪

A mixed number combines a whole number and a fraction, like $2\frac{1}{4}$.

Adding Mixed Numbers:

• Add the whole numbers together
• Add the fractions together
• Combine the results

Example: Anna Marie has $\frac{3}{4}$ of a cheese pizza and receives $\frac{3}{4}$ of a pepperoni pizza. How much pizza does she have?

$\frac{3}{4} + \frac{3}{4} = \frac{6}{4} = 1\frac{2}{4}$ or $1\frac{1}{2}$ pizzas

Fraction Bars: Draw rectangles divided into equal parts to show fractions visually

Number Lines: Mark fractions on a number line to see addition and subtraction

Fraction Circles: Use pie-shaped pieces to represent parts of a whole

1. Same Whole Rule: All fractions in a problem must refer to the same whole

   • ✅ Good: $\frac{1}{4}$ of a medium pizza + $\frac{1}{4}$ of a medium pizza
   • ❌ Bad: $\frac{1}{4}$ of a medium pizza + $\frac{1}{4}$ of a large pizza

2. Allowed Denominators: In Grade 4, you work with denominators of 2, 3, 4, 5, 6, 8, 10, 12, 16, and 100

3. No Simplifying Required: You don't need to reduce fractions to lowest terms yet

You can create stories for fraction equations! For example:

• Equation: $\frac{3}{10} + \frac{4}{10} = \frac{7}{10}$
• Story: "Maria ran $\frac{3}{10}$ of a mile in the morning and $\frac{4}{10}$ of a mile in the afternoon. How far did she run in total?"

🍰 Cooking: Adding $\frac{1}{2}$ cup flour + $\frac{1}{2}$ cup flour = $1$ cup flour

📊 Time: Spending $\frac{2}{6}$ hour on math + $\frac{3}{6}$ hour on reading = $\frac{5}{6}$ hour studying

🏃‍♀️ Sports: Running $\frac{3}{8}$ of a track + $\frac{2}{8}$ of a track = $\frac{5}{8}$ of a track

⚠️ Don't add denominators - Only add the numerators when denominators are the same

⚠️ Check that fractions refer to the same whole - You can't add parts of different-sized objects

⚠️ Use visual models when confused - Drawing helps you understand what's happening

Multiplying fractions by whole numbers opens up a whole new world of problem-solving! 🌟 You'll learn to work with groups of fractional parts, just like you work with groups of whole numbers.

When you multiply a fraction by a whole number, you're finding multiple groups of that fraction. Think of it like this:

• $4 × \frac{3}{5}$ means "four groups of three-fifths"
• $\frac{2}{3} × 8$ means "two-thirds of 8"

Type 1: Whole Number × Fraction

Example: Angelica walks $\frac{4}{5}$ of a mile every day. How far does she walk in 7 days?

$7 × \frac{4}{5} = \frac{7 × 4}{5} = \frac{28}{5} = 5\frac{3}{5}$ miles

Think of it as: $\frac{4}{5} + \frac{4}{5} + \frac{4}{5} + \frac{4}{5} + \frac{4}{5} + \frac{4}{5} + \frac{4}{5}$ 🚶‍♀️

Type 2: Fraction × Whole Number

Example: A butcher has 10 pounds of meat and sells $\frac{2}{3}$ of it. How many pounds does he sell?

$\frac{2}{3} × 10 = \frac{2 × 10}{3} = \frac{20}{3} = 6\frac{2}{3}$ pounds

This means "two-thirds of 10 pounds" 🥩

Fraction multiplication connects to what you already know about equal groups:

Whole Numbers: $4 × 3 = 3 + 3 + 3 + 3$ (four groups of 3)

Fractions: $4 × \frac{3}{5} = \frac{3}{5} + \frac{3}{5} + \frac{3}{5} + \frac{3}{5}$ (four groups of $\frac{3}{5}$)

Fraction Bars: Draw rectangles divided into fifths, then shade groups

Number Lines: Jump by fraction amounts to show repeated addition

Arrays: Use grids to show fractional parts of groups

When multiplying with mixed numbers, you can use the distributive property:

$2 × 6\frac{1}{3} = 2 × (6 + \frac{1}{3}) = (2 × 6) + (2 × \frac{1}{3}) = 12 + \frac{2}{3} = 12\frac{2}{3}$

🍪 Baking: A recipe calls for $\frac{2}{3}$ cup of sugar, but you want to make 4 batches: $4 × \frac{2}{3} = \frac{8}{3} = 2\frac{2}{3}$ cups of sugar

🏃‍♂️ Running: You've completed $\frac{3}{5}$ of a 10-mile run: $\frac{3}{5} × 10 = 6$ miles completed

🎨 Art: Making a project that needs $\frac{1}{4}$ yard of fabric, but you're making 6 projects: $6 × \frac{1}{4} = \frac{6}{4} = 1\frac{2}{4} = 1\frac{1}{2}$ yards needed

1. Identify what you're looking for - Total amount or partial amount?
2. Determine the multiplication type - Groups of fractions or fraction of a whole?
3. Set up the equation - Write it clearly with proper notation
4. Solve step by step - Show your work with fractions
5. Check with estimation - Does your answer make sense?
6. Write your final answer - Include units and context

Just like whole number units (1 cup, 2 miles), you can have fractional units:

• $\frac{1}{2}$ cup, $\frac{3}{4}$ mile, $\frac{2}{5}$ hour

When you multiply these fractional units, you get larger amounts:

• $3 × \frac{1}{2}$ cup = $1\frac{1}{2}$ cups
• $2 × \frac{3}{4}$ mile = $1\frac{1}{2}$ miles

Recipe Problems: Lorelei needs $\frac{2}{3}$ cup of sugar for 4 batches of cookies and $\frac{1}{3}$ cup for 8 smoothies:

• Cookies: $4 × \frac{2}{3} = \frac{8}{3} = 2\frac{2}{3}$ cups
• Smoothies: $8 × \frac{1}{3} = \frac{8}{3} = 2\frac{2}{3}$ cups
• Total: $2\frac{2}{3} + 2\frac{2}{3} = 5\frac{1}{3}$ cups

Measurement Problems: Ramon needs $\frac{1}{3}$ of a recipe that calls for 2 cups of sugar: $\frac{1}{3} × 2 = \frac{2}{3}$ cup of sugar

⚠️ Don't multiply both numerator and denominator by the whole number - Only multiply the numerator

⚠️ Remember that fractions are numbers too - They can be multiplied just like whole numbers

⚠️ Use visual models when confused - Drawing helps you understand what's happening

⚠️ Check that your answer makes sense - If you're finding part of something, your answer should be smaller

The equal sign is one of the most important symbols in mathematics, but it's often misunderstood! 🤔 Let's explore what it really means and how to determine if equations are true or false.

Many students think the equal sign means "the answer is" or "calculate this." But the equal sign actually means "the same as" or "is equal to."

Think of it like a balance scale ⚖️:

• The left side must have the same value as the right side
• If both sides have the same value, the equation is true
• If the sides have different values, the equation is false

True Equations:

• $7 + 5 = 12$ (both sides equal 12)
• $6 × 4 = 24$ (both sides equal 24)
• $15 - 8 = 14 - 7$ (both sides equal 7)
• $3 × 8 = 4 × 6$ (both sides equal 24)

False Equations:

• $9 + 4 = 12$ (left side = 13, right side = 12)
• $5 × 6 = 35$ (left side = 30, right side = 35)
• $20 ÷ 4 = 6$ (left side = 5, right side = 6)

Strategy 1: Calculate Both Sides

For the equation $86 + 58 = 144 ÷ 12$:

• Left side: $86 + 58 = 144$
• Right side: $144 ÷ 12 = 12$
• Since $144 ≠ 12$, the equation is false

Strategy 2: Use Comparative Thinking

Sometimes you don't need to calculate everything! For $32 ÷ 8 = 32 - 8 - 8 - 8 - 8$:

• Left side: $32 ÷ 8 = 4$
• Right side: You can see this subtracts 8 four times from 32, which gives 0
• Since $4 ≠ 0$, the equation is false

Strategy 3: Use Number Properties

For $6 × 7 = 7 × 6$:

• You know multiplication is commutative (order doesn't matter)
• Both sides equal 42, so it's true

Unlike simple arithmetic, equations can have operations on both sides:

• $24 ÷ 6 = 2 × 2$ (both sides equal 4, so true)
• $5 + 7 = 15 - 3$ (both sides equal 12, so true)
• $8 × 3 = 30 - 6$ (left = 24, right = 24, so true)

You can create your own true equations! Using the numbers 3, 5, 6, 10, create: $3 × 10 = 5 × 6$ (both sides equal 30)

Or: $10 + 5 = 6 + 3 + 6$ (both sides equal 15)

Understanding equality helps you:

• Solve more complex problems in higher grades
• Check your work by seeing if both sides balance
• Understand algebra where you'll work with unknown values
• Think flexibly about numbers and operations

Addition and Subtraction:

• $45 + 23 = 68$ (true)
• $92 - 37 = 55$ (true)
• $46 + 28 = 70 + 4$ (both equal 74, true)

Multiplication and Division:

• $7 × 8 = 56$ (true)
• $72 ÷ 9 = 8$ (true)
• $4 × 12 = 6 × 8$ (both equal 48, true)

$25 + 15 = 5 × 8$

• Left: $25 + 15 = 40$
• Right: $5 × 8 = 40$
• Both sides equal 40, so it's true!

⚠️ Don't assume the equal sign means "find the answer" - It means both sides are the same value

⚠️ Don't only look at one side - Always check both sides of the equation

⚠️ Don't rush - Take time to calculate or think through each side carefully

⚠️ Remember operation order - Follow the order of operations when evaluating

1. Look at both sides of the equation
2. Calculate or estimate the value of each side
3. Compare the values - Are they the same?
4. Determine if the equation is true or false
5. Explain your reasoning - Why is it true or false?

Equations represent balance in real life:

• Money: $\$5 + \$3 = \$8$ (total money)
• Time: $2 \text{ hours} + 30 \text{ minutes} = 150 \text{ minutes}$
• Objects: $3 \text{ boxes} × 4 \text{ items} = 12 \text{ items}$

When these relationships are correct, we have true equations!

Writing equations is like translating between English and mathematics! 📝 You'll learn to represent real-world situations with equations and solve for unknown values using letters.

In algebra, we use letters (called variables) to represent unknown numbers:

• $x$ = unknown number of apples
• $t$ = unknown ticket price
• $c$ = unknown number of cars
• $n$ = unknown number of students

These letters act as placeholders for the values we're trying to find! 🔍

Pattern: (Number of groups) × (Size of each group) = Total

Example 1: Movie tickets cost $\$12$ each. If 8 tickets cost $\$96$, what's the price per ticket?

Equation: $8 × t = 96$

Solution: $t = 96 ÷ 8 = 12$

So each ticket costs $\$12$! 🎬

Example 2: A typical Dalmatian weighs 54 pounds and a Yorkshire terrier weighs 9 pounds. How many times heavier is the Dalmatian?

Equation: $9 × n = 54$

Solution: $n = 54 ÷ 9 = 6$

The Dalmatian weighs 6 times as much! 🐕

Pattern: Total ÷ (Number of groups) = Size of each group

Example: Shernice has 84 comic books, which is 12 times as many as Cindy. How many comic books does Cindy have?

Equation: $84 = 12 × C$

Solution: $C = 84 ÷ 12 = 7$

Cindy has 7 comic books! 📚

The unknown can appear anywhere in the equation:

Unknown at the end: $6 × 7 = n$ Unknown in the middle: $6 × n = 42$ Unknown at the beginning: $n × 7 = 42$

All of these can be solved using the relationship between multiplication and division!

Multiplication and division are inverse operations - they undo each other:

• If $a × b = c$, then $c ÷ a = b$ and $c ÷ b = a$
• If $24 ÷ 6 = 4$, then $6 × 4 = 24$ and $4 × 6 = 24$

This relationship helps you solve equations! 🔄

These problems use phrases like "times as many" or "times as much":

Example: Enrique has 63 baseball cards, which is 9 times as many as Damion. How many does Damion have?

Translation: Damion's cards × 9 = Enrique's cards

Equation: $d × 9 = 63$

Solution: $d = 63 ÷ 9 = 7$

Damion has 7 cards! ⚾

Bar Models help you visualize problems:

For "63 is 9 times as many as unknown":

Damion:  [___]
Enrique: [___][___][___][___][___][___][___][___][___]

Each box represents the same amount, so $9 × \text{one box} = 63$

1. Read the problem carefully - What are you trying to find?
2. Identify the relationship - Is it multiplication or division?
3. Choose a letter for the unknown value
4. Write the equation - Translate the words into math
5. Solve the equation - Use inverse operations
6. Check your answer - Does it make sense?
7. Write your final answer - Include units and context

🏃‍♀️ Distance: Sabrina hiked 48 miles, which is 8 times more than Andre. How far did Andre hike?

Equation: $8 × a = 48$ Solution: $a = 48 ÷ 8 = 6$ miles

🏫 School: There are 144 students in the cafeteria sitting at tables of 12. How many tables are there?

Equation: $12 × t = 144$ Solution: $t = 144 ÷ 12 = 12$ tables

🎁 Shopping: You have $\$60$ to buy gifts that cost $\$15$ each. How many gifts can you buy?

Equation: $15 × g = 60$ Solution: $g = 60 ÷ 15 = 4$ gifts

Factor × Factor = Product

• $7 × ? = 56$ (Find the missing factor)
• $? × 8 = 72$ (Find the missing factor)
• $6 × 9 = ?$ (Find the product)

Dividend ÷ Divisor = Quotient

• $84 ÷ ? = 12$ (Find the divisor)
• $? ÷ 7 = 9$ (Find the dividend)
• $63 ÷ 9 = ?$ (Find the quotient)

"Times as many" → Multiplication

• "5 times as many" means multiply by 5
• "3 times as much" means multiply by 3

"How many groups" → Division

• "How many groups of 4?" means divide by 4
• "How many sets of 6?" means divide by 6

"Each" or "per" → Division or multiplication

• "Each person gets..." often means division
• "Cost per item" means division

Substitution Method:

1. Take your answer and substitute it back into the original equation
2. Check if both sides are equal
3. If yes, your answer is correct!

Example: If $x = 7$ for $5 × x = 35$ Check: $5 × 7 = 35$ ✓

⚠️ Don't confuse the unknown's position - Pay attention to what the problem is asking

⚠️ Don't forget to use the inverse operation - If you multiply, divide to solve

⚠️ Don't skip the checking step - Always verify your answer makes sense

⚠️ Don't forget units - Include dollars, miles, students, etc. in your final answer

Every number has a unique set of factors that tell us about its mathematical properties! 🔢 Understanding factors helps you work with numbers more effectively and prepares you for more advanced math concepts.

Factors are numbers that divide evenly into another number with no remainder. They come in factor pairs - two numbers that multiply together to give the original number.

Example: For the number 12

• $1 × 12 = 12$ (factor pair: 1 and 12)
• $2 × 6 = 12$ (factor pair: 2 and 6)
• $3 × 4 = 12$ (factor pair: 3 and 4)

So 12 has factors: 1, 2, 3, 4, 6, 12

Arrays (rectangular arrangements) help visualize factors:

For 12 objects:

1 × 12: ●●●●●●●●●●●●
2 × 6:  ●●●●●●
        ●●●●●●
3 × 4:  ●●●●
        ●●●●
        ●●●●

Each arrangement shows a different factor pair! 📐

These rules help you quickly identify factors:

Divisible by 2: Number is even (ends in 0, 2, 4, 6, 8)

• 24, 46, 108 are divisible by 2

Divisible by 3: Sum of digits is divisible by 3

• 147: $1 + 4 + 7 = 12$, and 12 ÷ 3 = 4, so 147 is divisible by 3

Divisible by 4: Last two digits form a number divisible by 4

• 1,236: 36 ÷ 4 = 9, so 1,236 is divisible by 4

Divisible by 5: Ends in 0 or 5

• 45, 120, 335 are divisible by 5

Divisible by 6: Divisible by both 2 and 3

• 42: even (÷2) and $4 + 2 = 6$ (÷3), so divisible by 6

Divisible by 9: Sum of digits is divisible by 9

• 729: $7 + 2 + 9 = 18$, and 18 ÷ 9 = 2, so 729 is divisible by 9

A prime number has exactly two factors: 1 and itself.

Examples of prime numbers:

• 2 (factors: 1, 2) - the only even prime! 🌟
• 3 (factors: 1, 3)
• 5 (factors: 1, 5)
• 7 (factors: 1, 7)
• 11 (factors: 1, 11)
• 13 (factors: 1, 13)

First 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

A composite number has more than two factors.

Examples of composite numbers:

• 4 (factors: 1, 2, 4)
• 6 (factors: 1, 2, 3, 6)
• 8 (factors: 1, 2, 4, 8)
• 9 (factors: 1, 3, 9)
• 10 (factors: 1, 2, 5, 10)

0 and 1 are neither prime nor composite!

• 0: Every number is a factor of 0, so it has infinitely many factors
• 1: Only has one factor (itself), but prime numbers need exactly two factors

Example: Finding factors of 48

1. Start with 1: $1 × 48 = 48$ ✓
2. Try 2: $2 × 24 = 48$ ✓
3. Try 3: $3 × 16 = 48$ ✓
4. Try 4: $4 × 12 = 48$ ✓
5. Try 5: $48 ÷ 5 = 9.6$ ✗ (not a whole number)
6. Try 6: $6 × 8 = 48$ ✓
7. Try 7: $48 ÷ 7 = 6.86...$ ✗
8. Try 8: Already found (8 × 6)

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Since 48 has more than two factors, it's composite. 🔢

Let's find all factors and classify each number:

20: 1, 2, 4, 5, 10, 20 (composite) 21: 1, 3, 7, 21 (composite) 22: 1, 2, 11, 22 (composite) 23: 1, 23 (prime) ⭐ 24: 1, 2, 3, 4, 6, 8, 12, 24 (composite) 25: 1, 5, 25 (composite) 26: 1, 2, 13, 26 (composite) 27: 1, 3, 9, 27 (composite) 28: 1, 2, 4, 7, 14, 28 (composite) 29: 1, 29 (prime) ⭐ 30: 1, 2, 3, 5, 6, 10, 15, 30 (composite)

Greatest prime in this range: 29 Least prime in this range: 23

Let's examine all the statements about 84:

Finding factors of 84:

• $1 × 84 = 84$
• $2 × 42 = 84$
• $3 × 28 = 84$
• $4 × 21 = 84$
• $6 × 14 = 84$
• $7 × 12 = 84$

Factors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Factor pairs: (1,84), (2,42), (3,28), (4,21), (6,14), (7,12)

Analysis:

• 42 is a factor of 84 ✓
• 84 is composite (has more than 2 factors) ✓
• 84 has 6 distinct factor pairs ✗ (statement says 4)
• Prime factors would be 2, 3, 7 (not 2, 6, 7) ✗
• $84 = 2 × 2 × 3 × 7 = 2² × 3 × 7$ ✓

Skip counting helps check factors:

• Is 32 divisible by 4? Count: 4, 8, 12, 16, 20, 24, 28, 32 ✓
• Is 35 divisible by 7? Count: 7, 14, 21, 28, 35 ✓

Multiplication facts help identify factors:

• Know that $6 × 7 = 42$? Then 6 and 7 are factors of 42!

🏫 Arranging desks: 24 desks can be arranged in arrays of:

• 1 × 24, 2 × 12, 3 × 8, 4 × 6

🎁 Packaging: 36 items can be packaged in:

• 1 box of 36, 2 boxes of 18, 3 boxes of 12, 4 boxes of 9, 6 boxes of 6

🏃‍♀️ Teams: 20 students can form teams of:

• 1 team of 20, 2 teams of 10, 4 teams of 5

⚠️ Don't think 1 is prime - It only has one factor, but primes need exactly two

⚠️ Don't assume all odd numbers are prime - 9, 15, 21, 25, 27 are odd but composite

⚠️ Don't think larger numbers always have more factors - 17 has fewer factors than 12

⚠️ Don't forget to check systematically - Use divisibility rules and arrays to be thorough

Patterns are everywhere in mathematics! 🎯 Learning to create, describe, and extend numerical patterns helps you understand relationships between numbers and strengthens your operation skills.

A numerical pattern is a sequence of numbers that follows a specific rule. The rule determines how you get from one number to the next.

Examples:

• Start at 5, add 3: 5, 8, 11, 14, 17, 20...
• Start at 48, subtract 6: 48, 42, 36, 30, 24, 18...
• Start at 2, multiply by 3: 2, 6, 18, 54, 162...

The rule tells you what operation to perform and by what number:

Addition Rules:

• "Add 7" means: 3, 10, 17, 24, 31...
• "Add 15" means: 8, 23, 38, 53, 68...

Subtraction Rules:

• "Subtract 4" means: 50, 46, 42, 38, 34...
• "Subtract 9" means: 72, 63, 54, 45, 36...

Multiplication Rules:

• "Multiply by 2" means: 3, 6, 12, 24, 48...
• "Multiply by 4" means: 1, 4, 16, 64, 256...

Example 1: Generate 4 numbers starting at 5, rule "add 14"

• Start: 5
• Add 14: 5 + 14 = 19
• Add 14: 19 + 14 = 33
• Add 14: 33 + 14 = 47
• Pattern: 5, 19, 33, 47

Example 2: Generate 5 numbers starting at 100, rule "subtract 12"

• Start: 100
• Subtract 12: 100 - 12 = 88
• Subtract 12: 88 - 12 = 76
• Subtract 12: 76 - 12 = 64
• Subtract 12: 64 - 12 = 52
• Pattern: 100, 88, 76, 64, 52

When you see a pattern, you can describe it by finding the rule:

Pattern: 6, 10, 14, 18, 22... Rule: Start at 6, add 4 Description: "Each term increases by 4"

Pattern: 81, 72, 63, 54, 45... Rule: Start at 81, subtract 9 Description: "Each term decreases by 9"

Example: The pattern 6, 10, 14, 18, 22 follows "add 4"

To extend it:

• 22 + 4 = 26
• 26 + 4 = 30
• 30 + 4 = 34
• Extended pattern: 6, 10, 14, 18, 22, 26, 30, 34...

Example: Pattern starts at 6, rule "add 4": 6, 10, 14, 18, 22, 26, 30...

Which numbers belong?

• 8? No (not in the sequence)
• 14? Yes (third term)
• 16? No (not in the sequence)
• 30? Yes (seventh term)

Interesting Discovery: Starting number + rule = pattern type

Example: First term is odd, rule "add 13"

• Start: odd number
• Add 13 (odd): odd + odd = even
• Add 13 (odd): even + odd = odd
• Add 13 (odd): odd + odd = even
• Pattern: odd, even, odd, even...

The 4th term will always be even! 🎯

Money Patterns:

• Saving $\$5$ each week: $\$5, \$10, \$15, \$20, \$25$...
• Spending $\$8$ each day from $\$50$: $\$50, \$42, \$34, \$26, \$18$...

Time Patterns:

• Meeting every 3 hours starting at 9 AM: 9 AM, 12 PM, 3 PM, 6 PM...
• Countdown by 15 minutes: 60, 45, 30, 15, 0...

Measurement Patterns:

• Growing 2 inches each year: 48, 50, 52, 54, 56...
• Losing 3 pounds each month: 150, 147, 144, 141, 138...

Square Areas Pattern: Squares with side lengths 1, 2, 3, 4, 5...

Areas: $1², 2², 3², 4², 5²$ = 1, 4, 9, 16, 25...

Rule: The differences between consecutive squares follow the pattern:

• 4 - 1 = 3
• 9 - 4 = 5
• 16 - 9 = 7
• 25 - 16 = 9

Difference pattern: 3, 5, 7, 9... (add 2 each time)

Square Perimeters Pattern: Squares with side lengths 1, 2, 3, 4, 5...

Perimeters: $4×1, 4×2, 4×3, 4×4, 4×5$ = 4, 8, 12, 16, 20...

Rule: Start at 4, add 4 (or multiply position by 4)

Dot Patterns: For "start at 4, add 3":

Term 1: ●●●●
Term 2: ●●●●●●●
Term 3: ●●●●●●●●●●

Color Patterns: Use different colored tiles to show growing patterns visually

Patterns help you practice:

• Addition: 7, 14, 21, 28, 35... (adding 7 = 7 times table)
• Subtraction: 100, 92, 84, 76, 68... (subtracting 8)
• Multiplication: 3, 12, 48, 192... (multiplying by 4)

Strategy:

1. Identify the first term - What number does the pattern start with?
2. Find the rule - What operation and number create the pattern?
3. Apply the rule - Continue the pattern systematically
4. Check your work - Do the numbers follow the rule consistently?
5. Extend or analyze - Answer the specific question asked

Two-Step Patterns: Some patterns might have complex rules:

• Start at 1: 1, 3, 9, 27, 81... (multiply by 3)
• Start at 2: 2, 6, 18, 54, 162... (multiply by 3)

Growing Patterns:

• Start at 1, add 1: 1, 2, 3, 4, 5...
• Start at 1, add 2: 1, 3, 5, 7, 9...
• Start at 1, add 3: 1, 4, 7, 10, 13...

⚠️ Don't assume the rule without checking - Make sure it works for all given terms

⚠️ Don't make calculation errors - Use your operation facts accurately

⚠️ Don't forget to check if specific numbers belong - Test systematically

⚠️ Don't mix up the starting number - The first term is given, not calculated

Daily Practice:

• Create patterns using your age, birthday, or favorite number
• Look for patterns in house numbers, page numbers, or temperatures
• Practice skip counting as pattern recognition

Fact Fluency:

• Use multiplication patterns (2, 4, 6, 8...) to memorize times tables
• Use addition patterns (5, 10, 15, 20...) to practice adding

Real-World Connections:

• Track weekly allowance patterns
• Notice patterns in sports scores or statistics
• Find patterns in nature (flower petals, leaf arrangements) 🌸