🔢 Numbers and Comparing - Grade 3

🔵🔴 Even and Odd Numbers

What are Even Numbers?

Even numbers are whole numbers that can be divided by \(2\) exactly with no remainder!

Key Rules:

• Even numbers end in: \(0, 2, 4, 6,\) or \(8\)
• They can be divided by \(2\) with no remainder
• They can be grouped into pairs with none left over

Examples: \(2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 50, 100, 238\)

What are Odd Numbers?

Odd numbers are whole numbers that cannot be divided by \(2\) exactly - they always leave a remainder of \(1\)!

Key Rules:

• Odd numbers end in: \(1, 3, 5, 7,\) or \(9\)
• They leave a remainder of \(1\) when divided by \(2\)
• When grouped in pairs, there's always one left over

Examples: \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 51, 99, 237\)

Quick Test:

To check if a number is even or odd,
look at the LAST digit (ones place)!

➕➖✖️ Even and Odd Arithmetic Rules

➕ Addition Rules

First Number	Second Number	Sum	Example
Even	Even	Even	\(6 + 8 = 14\) (even)
Odd	Odd	Even	\(7 + 9 = 16\) (even)
Even	Odd	Odd	\(4 + 5 = 9\) (odd)
Odd	Even	Odd	\(3 + 6 = 9\) (odd)

➖ Subtraction Rules

First Number	Second Number	Difference	Example
Even	Even	Even	\(12 − 8 = 4\) (even)
Odd	Odd	Even	\(15 − 7 = 8\) (even)
Even	Odd	Odd	\(10 − 3 = 7\) (odd)
Odd	Even	Odd	\(13 − 6 = 7\) (odd)

✖️ Multiplication Rules

First Number	Second Number	Product	Example
Even	Even	Even	\(4 \times 6 = 24\) (even)
Odd	Odd	Odd	\(3 \times 5 = 15\) (odd)
Even	Odd	Even	\(6 \times 7 = 42\) (even)
Odd	Even	Even	\(5 \times 8 = 40\) (even)

⚡ Key Rule: Any number multiplied by an even number is ALWAYS even!

📝 Summary Formula

Even \(\pm\) Even = Even
Odd \(\pm\) Odd = Even
Even \(\pm\) Odd = Odd

Even \(\times\) Anything = Even
Odd \(\times\) Odd = Odd

🦘 Skip-Counting Puzzles

What is Skip-Counting?

Skip-counting means counting by a number other than \(1\) - jumping over numbers in a pattern!

\(\text{Pattern} = \text{Start} + n, + n, + n, + n...\)

Common Skip-Counting Patterns

Skip-Counting by 2s:

\(2, 4, 6, 8, 10, 12, 14, 16, 18, 20...\)

Skip-Counting by 5s:

\(5, 10, 15, 20, 25, 30, 35, 40, 45, 50...\)

Skip-Counting by 10s:

\(10, 20, 30, 40, 50, 60, 70, 80, 90, 100...\)

Skip-Counting by 3s:

\(3, 6, 9, 12, 15, 18, 21, 24, 27, 30...\)

Skip-Counting by 4s:

\(4, 8, 12, 16, 20, 24, 28, 32, 36, 40...\)

How to Solve Skip-Counting Puzzles

Find the pattern: Look at the difference between numbers
Check if it's the same: Subtract each number from the next
Apply the rule: Add the same amount to find missing numbers
Double-check: Make sure the pattern continues

Example: \(12, 15, 18, ?, 24\)
Pattern: \(+3\) each time
Missing number: \(18 + 3 = 21\) ✓

🔢 Number Sequences

What is a Number Sequence?

A number sequence is a list of numbers that follows a specific pattern or rule!

Each number in a sequence is called a term

Types of Sequences

1. Ascending Sequence (Increasing)

Numbers get larger as the sequence continues.

Examples:
• \(5, 10, 15, 20, 25\) (add \(5\))
• \(100, 200, 300, 400\) (add \(100\))
• \(3, 6, 9, 12, 15\) (add \(3\))

2. Descending Sequence (Decreasing)

Numbers get smaller as the sequence continues.

Examples:
• \(50, 45, 40, 35, 30\) (subtract \(5\))
• \(100, 90, 80, 70\) (subtract \(10\))
• \(30, 27, 24, 21\) (subtract \(3\))

Finding the Rule

\(\text{Next Term} = \text{Current Term} \pm \text{Rule}\)

Example Problem: Find the next two numbers: \(7, 14, 21, 28, ?, ?\)

Step 1: Find the difference: \(14 - 7 = 7\), \(21 - 14 = 7\), \(28 - 21 = 7\)
Step 2: The rule is "add \(7\)"
Step 3: Apply the rule: \(28 + 7 = 35\), \(35 + 7 = 42\)
Answer: \(35, 42\) ✓

🏆 Ordinal Numbers to 100th

What are Ordinal Numbers?

Ordinal numbers show the position or order of things in a list!

They answer the question: "Which position?"

Cardinal Numbers: Tell "how many" → \(1, 2, 3, 4, 5\)
Ordinal Numbers: Tell "which position" → 1st, 2nd, 3rd, 4th, 5th

Rules for Writing Ordinal Numbers

Rule 1: Numbers ending in 1 → use "st"

Examples: 1st, 21st, 31st, 41st, 51st, 61st, 71st, 81st, 91st
Exception: 11th (NOT 11st)

Rule 2: Numbers ending in 2 → use "nd"

Examples: 2nd, 22nd, 32nd, 42nd, 52nd, 62nd, 72nd, 82nd, 92nd
Exception: 12th (NOT 12nd)

Rule 3: Numbers ending in 3 → use "rd"

Examples: 3rd, 23rd, 33rd, 43rd, 53rd, 63rd, 73rd, 83rd, 93rd
Exception: 13th (NOT 13rd)

Rule 4: All other numbers → use "th"

Examples: 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th, 13th, 14th, 15th, 16th, 17th, 18th, 19th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, 90th, 100th

Quick Reference Chart

Number	Ordinal	In Words
\(1\)	1st	First
\(2\)	2nd	Second
\(3\)	3rd	Third
\(10\)	10th	Tenth
\(20\)	20th	Twentieth
\(50\)	50th	Fiftieth
\(100\)	100th	Hundredth

✍️ Write Numbers in Words

Place Value Chart

Thousands	Hundreds	Tens	Ones
\(5\)	\(3\)	\(4\)	\(7\)

\(5,347 = \) Five Thousand Three Hundred Forty-Seven

Steps to Write Numbers in Words

Break the number by place value
Write the thousands place first (if any)
Write the hundreds place (if any)
Write the tens and ones together
Connect with proper words (hundred, thousand, etc.)

Examples

1. \(234 = \) Two Hundred Thirty-Four
2. \(508 = \) Five Hundred Eight
3. \(1,056 = \) One Thousand Fifty-Six
4. \(3,421 = \) Three Thousand Four Hundred Twenty-One
5. \(9,000 = \) Nine Thousand

⚠️ Important Rules

Use hyphens for numbers 21-99 (twenty-one, thirty-five)
Use "and" only for decimals (not needed for whole numbers in US style)
Never use "and" between hundreds and tens
Always capitalize the first word

⚖️ Comparing Numbers

Comparison Symbols

\(>\) means "greater than" (bigger)
\(<\) means "less than" (smaller)
\(=\) means "equal to" (same)

💡 Trick: The symbol always points to the smaller number!

Steps to Compare Numbers

Count the digits: More digits = bigger number
If same digits, compare from left to right:
- Start with the thousands place
- Then hundreds place
- Then tens place
- Finally ones place
Stop when you find a difference
Use the correct symbol

Examples

Example 1: Different Number of Digits

Compare: \(234\) and \(1,567\)

\(234\) has \(3\) digits
\(1,567\) has \(4\) digits
Answer: \(234 < 1,567\) ✓

Example 2: Same Number of Digits

Compare: \(3,456\) and \(3,489\)

Thousands: \(3 = 3\) (same)
Hundreds: \(4 = 4\) (same)
Tens: \(5 < 8\) (different!) ← Stop here
Answer: \(3,456 < 3,489\) ✓

Example 3: Equal Numbers

Compare: \(6,782\) and \(6,782\)

All digits are the same!
Answer: \(6,782 = 6,782\) ✓

🏅 Which Number is Greatest/Least?

Definitions

Greatest: The biggest or largest number
Least: The smallest number

How to Find Greatest/Least

Look at number of digits first
- Most digits = greatest
- Fewest digits = least
If all have same digits, compare from left
Compare each place value until you find the answer

Example Problem

Find the greatest and least numbers:
\(234, 1,456, 89, 5,678, 567\)

Step 1: Count digits in each number:
• \(234\) → \(3\) digits
• \(1,456\) → \(4\) digits
• \(89\) → \(2\) digits ← Least!
• \(5,678\) → \(4\) digits
• \(567\) → \(3\) digits

Step 2: Find greatest (compare 4-digit numbers):
\(1,456\) vs \(5,678\)
Thousands place: \(1 < 5\)
So \(5,678\) is greatest! ✓

Answer:
Greatest: \(5,678\)
Least: \(89\)

🔢 Put Numbers in Order

Types of Ordering

1. Ascending Order (Least to Greatest)

Arrange numbers from smallest to largest

Example: \(23, 67, 89, 234, 567\) ✓

2. Descending Order (Greatest to Least)

Arrange numbers from largest to smallest

Example: \(567, 234, 89, 67, 23\) ✓

Steps to Order Numbers

Compare the numbers using place value
Find the smallest (or largest) first
Continue comparing the remaining numbers
Write them in order from least to greatest (or vice versa)
Check your answer!

Complete Example

Problem: Put these numbers in ascending order:
\(456, 89, 1,234, 567, 45\)

Step 1: Identify digit counts:
• \(45\) → \(2\) digits
• \(89\) → \(2\) digits
• \(456\) → \(3\) digits
• \(567\) → \(3\) digits
• \(1,234\) → \(4\) digits

Step 2: Order by digit count first, then by value:
2-digit: \(45 < 89\)
3-digit: \(456 < 567\)
4-digit: \(1,234\)

Final Answer (Ascending):
\(45, 89, 456, 567, 1,234\) ✓